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Properties

Label 4-8954e2-1.1-c1e2-0-6
Degree $4$
Conductor $80174116$
Sign $1$
Analytic cond. $5111.97$
Root an. cond. $8.45565$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s + 2·5-s + 4·7-s + 4·8-s − 4·9-s + 4·10-s + 4·13-s + 8·14-s + 5·16-s + 2·17-s − 8·18-s + 8·19-s + 6·20-s + 4·23-s − 5·25-s + 8·26-s + 12·28-s + 12·29-s + 4·31-s + 6·32-s + 4·34-s + 8·35-s − 12·36-s + 2·37-s + 16·38-s + 8·40-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 0.894·5-s + 1.51·7-s + 1.41·8-s − 4/3·9-s + 1.26·10-s + 1.10·13-s + 2.13·14-s + 5/4·16-s + 0.485·17-s − 1.88·18-s + 1.83·19-s + 1.34·20-s + 0.834·23-s − 25-s + 1.56·26-s + 2.26·28-s + 2.22·29-s + 0.718·31-s + 1.06·32-s + 0.685·34-s + 1.35·35-s − 2·36-s + 0.328·37-s + 2.59·38-s + 1.26·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80174116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80174116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(80174116\)    =    \(2^{2} \cdot 11^{4} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(5111.97\)
Root analytic conductor: \(8.45565\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 80174116,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(19.37786068\)
\(L(\frac12)\) \(\approx\) \(19.37786068\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad2$C_1$ \( ( 1 - T )^{2} \)
11 \( 1 \)
37$C_1$ \( ( 1 - T )^{2} \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.3.a_e
5$D_{4}$ \( 1 - 2 T + 9 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.5.ac_j
7$D_{4}$ \( 1 - 4 T + 16 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.7.ae_q
13$D_{4}$ \( 1 - 4 T + 28 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.13.ae_bc
17$D_{4}$ \( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.17.ac_bb
19$D_{4}$ \( 1 - 8 T + 52 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.19.ai_ca
23$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.23.ae_s
29$D_{4}$ \( 1 - 12 T + 92 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.29.am_do
31$D_{4}$ \( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.31.ae_cg
41$D_{4}$ \( 1 - 20 T + 180 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.41.au_gy
43$D_{4}$ \( 1 + 4 T + 40 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.43.e_bo
47$D_{4}$ \( 1 + 6 T + 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.47.g_cb
53$D_{4}$ \( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.53.ai_dm
59$D_{4}$ \( 1 + 12 T + 146 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.59.m_fq
61$D_{4}$ \( 1 - 8 T + 40 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.61.ai_bo
67$D_{4}$ \( 1 - 8 T + 100 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.67.ai_dw
71$D_{4}$ \( 1 + 10 T + 165 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.71.k_gj
73$C_2^2$ \( 1 - 96 T^{2} + p^{2} T^{4} \) 2.73.a_ads
79$D_{4}$ \( 1 + 10 T + 181 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.79.k_gz
83$D_{4}$ \( 1 - 6 T + 167 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.83.ag_gl
89$D_{4}$ \( 1 + 16 T + 234 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.89.q_ja
97$D_{4}$ \( 1 + 4 T - 44 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.97.e_abs
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.80925525769786243835363356909, −7.57611338526197272951555818503, −7.25044458428147615273136380071, −6.69249334041594480834992678928, −6.24882931997137271280676308632, −6.15323249654228310477938042582, −5.71269521873128975844955524148, −5.49328504990412725501150440732, −5.28403657287129199853145125017, −4.75261533658739044458547638389, −4.49633605200516637795364396146, −4.27775971049550099917551007214, −3.51831683667119937082850212246, −3.27921956689601571654134061447, −2.89900162880682760555262060592, −2.58597956412012933648534270839, −2.10311493857618642765811114004, −1.61587263987191982570434848209, −0.975846593707394018645990163976, −0.962462709574703662001688144797, 0.962462709574703662001688144797, 0.975846593707394018645990163976, 1.61587263987191982570434848209, 2.10311493857618642765811114004, 2.58597956412012933648534270839, 2.89900162880682760555262060592, 3.27921956689601571654134061447, 3.51831683667119937082850212246, 4.27775971049550099917551007214, 4.49633605200516637795364396146, 4.75261533658739044458547638389, 5.28403657287129199853145125017, 5.49328504990412725501150440732, 5.71269521873128975844955524148, 6.15323249654228310477938042582, 6.24882931997137271280676308632, 6.69249334041594480834992678928, 7.25044458428147615273136380071, 7.57611338526197272951555818503, 7.80925525769786243835363356909

Graph of the $Z$-function along the critical line