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Properties

Label 4-2376e2-1.1-c1e2-0-18
Degree $4$
Conductor $5645376$
Sign $-1$
Analytic cond. $359.954$
Root an. cond. $4.35573$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 3·5-s + 9-s − 3·11-s + 3·15-s + 3·23-s − 3·25-s + 27-s + 31-s − 3·33-s − 10·37-s + 3·45-s + 12·47-s + 4·49-s + 9·53-s − 9·55-s − 15·59-s + 11·67-s + 3·69-s − 6·71-s − 3·75-s + 81-s − 21·89-s + 93-s − 18·97-s − 3·99-s + 19·103-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.34·5-s + 1/3·9-s − 0.904·11-s + 0.774·15-s + 0.625·23-s − 3/5·25-s + 0.192·27-s + 0.179·31-s − 0.522·33-s − 1.64·37-s + 0.447·45-s + 1.75·47-s + 4/7·49-s + 1.23·53-s − 1.21·55-s − 1.95·59-s + 1.34·67-s + 0.361·69-s − 0.712·71-s − 0.346·75-s + 1/9·81-s − 2.22·89-s + 0.103·93-s − 1.82·97-s − 0.301·99-s + 1.87·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5645376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5645376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5645376\)    =    \(2^{6} \cdot 3^{6} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(359.954\)
Root analytic conductor: \(4.35573\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 5645376,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( 1 - T \)
11$C_2$ \( 1 + 3 T + p T^{2} \)
good5$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.5.ad_m
7$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.7.a_ae
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.17.a_q
19$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.19.a_au
23$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.23.ad_bq
29$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.29.a_s
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.31.ab_g
37$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.37.k_dm
41$C_2^2$ \( 1 - 72 T^{2} + p^{2} T^{4} \) 2.41.a_acu
43$C_2^2$ \( 1 - 28 T^{2} + p^{2} T^{4} \) 2.43.a_abc
47$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.47.am_fa
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.53.aj_eu
59$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.p_fy
61$C_2^2$ \( 1 + 47 T^{2} + p^{2} T^{4} \) 2.61.a_bv
67$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.67.al_ee
71$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.g_cs
73$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.73.a_abi
79$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.79.a_acg
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.89.v_ki
97$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.97.s_kg
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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

−7.14547895594713385829062578639, −6.72213087166247846870984094152, −6.15627280481867887522258609478, −5.78064725054279014539322722716, −5.49647395779911242527976464230, −5.14987176164258128879219966947, −4.62232315274450909207258892462, −4.06298399860180408714556406019, −3.68624778241969692946322060929, −3.04761578591271667947863895334, −2.56676658183254362245340485512, −2.26651966422633427870741626157, −1.69348242571818390203818955124, −1.13312061394671023937564048920, 0, 1.13312061394671023937564048920, 1.69348242571818390203818955124, 2.26651966422633427870741626157, 2.56676658183254362245340485512, 3.04761578591271667947863895334, 3.68624778241969692946322060929, 4.06298399860180408714556406019, 4.62232315274450909207258892462, 5.14987176164258128879219966947, 5.49647395779911242527976464230, 5.78064725054279014539322722716, 6.15627280481867887522258609478, 6.72213087166247846870984094152, 7.14547895594713385829062578639

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