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Properties

Label 4-1812608-1.1-c1e2-0-5
Degree $4$
Conductor $1812608$
Sign $1$
Analytic cond. $115.573$
Root an. cond. $3.27879$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 4·7-s − 8-s + 2·9-s − 4·14-s + 16-s − 2·18-s + 10·25-s + 4·28-s − 32-s + 2·36-s + 9·49-s − 10·50-s − 4·56-s + 8·63-s + 64-s − 2·72-s − 16·79-s − 5·81-s − 9·98-s + 10·100-s + 4·112-s + 12·113-s + 14·121-s − 8·126-s + 127-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 1.51·7-s − 0.353·8-s + 2/3·9-s − 1.06·14-s + 1/4·16-s − 0.471·18-s + 2·25-s + 0.755·28-s − 0.176·32-s + 1/3·36-s + 9/7·49-s − 1.41·50-s − 0.534·56-s + 1.00·63-s + 1/8·64-s − 0.235·72-s − 1.80·79-s − 5/9·81-s − 0.909·98-s + 100-s + 0.377·112-s + 1.12·113-s + 1.27·121-s − 0.712·126-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1812608 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1812608 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1812608\)    =    \(2^{7} \cdot 7^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(115.573\)
Root analytic conductor: \(3.27879\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1812608,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.239837209\)
\(L(\frac12)\) \(\approx\) \(2.239837209\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
17$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.3.a_ac
5$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.5.a_ak
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.a_ao
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.13.a_aw
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.19.a_aw
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.37.a_cg
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.a_w
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.59.a_aeo
61$C_2^2$ \( 1 - 106 T^{2} + p^{2} T^{4} \) 2.61.a_aec
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.67.a_cs
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.73.a_fm
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.79.q_io
83$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.83.a_agk
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.a_fm
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.97.a_ac
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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

−7.80779099796685648877747579650, −7.39365854227463027857068672397, −7.16816596018324611182838551848, −6.60956708114938405912386299165, −6.27671765160795091222272193282, −5.53097302440651000338695555060, −5.24492263244224630888475633184, −4.70745779614324596405408448858, −4.37008435137565935711272996584, −3.83794294179202234322179005510, −3.03550655537649844582766040359, −2.64152822917227112231747024781, −1.82222654636608010237362416442, −1.44689560592986068146993246480, −0.75292915042365455186689429445, 0.75292915042365455186689429445, 1.44689560592986068146993246480, 1.82222654636608010237362416442, 2.64152822917227112231747024781, 3.03550655537649844582766040359, 3.83794294179202234322179005510, 4.37008435137565935711272996584, 4.70745779614324596405408448858, 5.24492263244224630888475633184, 5.53097302440651000338695555060, 6.27671765160795091222272193282, 6.60956708114938405912386299165, 7.16816596018324611182838551848, 7.39365854227463027857068672397, 7.80779099796685648877747579650

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