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Properties

Label 4-19e4-1.1-c1e2-0-2
Degree $4$
Conductor $130321$
Sign $1$
Analytic cond. $8.30937$
Root an. cond. $1.69782$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3·3-s − 2·4-s + 2·5-s + 3·6-s + 6·7-s + 3·8-s + 2·9-s − 2·10-s − 11-s + 6·12-s + 2·13-s − 6·14-s − 6·15-s + 16-s + 6·17-s − 2·18-s − 4·20-s − 18·21-s + 22-s + 13·23-s − 9·24-s − 2·25-s − 2·26-s + 6·27-s − 12·28-s + 5·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.73·3-s − 4-s + 0.894·5-s + 1.22·6-s + 2.26·7-s + 1.06·8-s + 2/3·9-s − 0.632·10-s − 0.301·11-s + 1.73·12-s + 0.554·13-s − 1.60·14-s − 1.54·15-s + 1/4·16-s + 1.45·17-s − 0.471·18-s − 0.894·20-s − 3.92·21-s + 0.213·22-s + 2.71·23-s − 1.83·24-s − 2/5·25-s − 0.392·26-s + 1.15·27-s − 2.26·28-s + 0.928·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130321 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130321 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(130321\)    =    \(19^{4}\)
Sign: $1$
Analytic conductor: \(8.30937\)
Root analytic conductor: \(1.69782\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 130321,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7555529873\)
\(L(\frac12)\) \(\approx\) \(0.7555529873\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad19 \( 1 \)
good2$D_{4}$ \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) 2.2.b_d
3$D_{4}$ \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) 2.3.d_h
5$D_{4}$ \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.5.ac_g
7$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.7.ag_x
11$C_4$ \( 1 + T + 21 T^{2} + p T^{3} + p^{2} T^{4} \) 2.11.b_v
13$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.13.ac_bb
17$D_{4}$ \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.17.ag_bm
23$D_{4}$ \( 1 - 13 T + 87 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.23.an_dj
29$D_{4}$ \( 1 - 5 T + 63 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.29.af_cl
31$C_4$ \( 1 - 11 T + 81 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.31.al_dd
37$D_{4}$ \( 1 + 11 T + 93 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.37.l_dp
41$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.41.ag_dn
43$D_{4}$ \( 1 + 7 T + 87 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.43.h_dj
47$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.47.ag_dz
53$D_{4}$ \( 1 + 3 T + 47 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.53.d_bv
59$D_{4}$ \( 1 - 15 T + 113 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.59.ap_ej
61$D_{4}$ \( 1 + 16 T + 181 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.61.q_gz
67$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.67.ao_hb
71$D_{4}$ \( 1 - 6 T + 131 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.71.ag_fb
73$D_{4}$ \( 1 - 8 T + 117 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.73.ai_en
79$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.79.a_aw
83$D_{4}$ \( 1 - 8 T + 162 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.83.ai_gg
89$D_{4}$ \( 1 - 20 T + 273 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.89.au_kn
97$D_{4}$ \( 1 + 21 T + 293 T^{2} + 21 p T^{3} + p^{2} T^{4} \) 2.97.v_lh
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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

−11.57882736085552198134522935028, −10.96231411824894405903996298505, −10.84080307822723905981779615242, −10.50075657344981802187925330318, −9.798697706800423014357455690261, −9.538724579617100791959127448412, −8.676969917827838407562273032034, −8.555598155759025452232880594111, −8.082805617875589722138498482881, −7.62720436963227164702038916189, −6.77840075929955903821020585333, −6.33424279004026911203339261188, −5.57168648237408996686983597590, −5.25500223375127053982206208771, −4.82422082905583063395826771604, −4.81694218467493453071981579001, −3.62294796890553037716187074310, −2.55065591875919869390471850832, −1.25916730779689270987259748533, −0.963535653843775763640796123360, 0.963535653843775763640796123360, 1.25916730779689270987259748533, 2.55065591875919869390471850832, 3.62294796890553037716187074310, 4.81694218467493453071981579001, 4.82422082905583063395826771604, 5.25500223375127053982206208771, 5.57168648237408996686983597590, 6.33424279004026911203339261188, 6.77840075929955903821020585333, 7.62720436963227164702038916189, 8.082805617875589722138498482881, 8.555598155759025452232880594111, 8.676969917827838407562273032034, 9.538724579617100791959127448412, 9.798697706800423014357455690261, 10.50075657344981802187925330318, 10.84080307822723905981779615242, 10.96231411824894405903996298505, 11.57882736085552198134522935028

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