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Properties

Label 4-15424-1.1-c1e2-0-1
Degree $4$
Conductor $15424$
Sign $-1$
Analytic cond. $0.983447$
Root an. cond. $0.995835$
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  − 2-s − 4-s − 2·7-s + 3·8-s − 4·9-s + 2·14-s − 16-s − 3·17-s + 4·18-s − 2·23-s − 3·25-s + 2·28-s − 6·31-s − 5·32-s + 3·34-s + 4·36-s − 12·41-s + 2·46-s + 4·47-s − 11·49-s + 3·50-s − 6·56-s + 6·62-s + 8·63-s + 7·64-s + 3·68-s + 3·71-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.755·7-s + 1.06·8-s − 4/3·9-s + 0.534·14-s − 1/4·16-s − 0.727·17-s + 0.942·18-s − 0.417·23-s − 3/5·25-s + 0.377·28-s − 1.07·31-s − 0.883·32-s + 0.514·34-s + 2/3·36-s − 1.87·41-s + 0.294·46-s + 0.583·47-s − 1.57·49-s + 0.424·50-s − 0.801·56-s + 0.762·62-s + 1.00·63-s + 7/8·64-s + 0.363·68-s + 0.356·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(15424\)    =    \(2^{6} \cdot 241\)
Sign: $-1$
Analytic conductor: \(0.983447\)
Root analytic conductor: \(0.995835\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 15424,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + T + p T^{2} \)
241$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 5 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.3.a_e
5$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \) 2.5.a_d
7$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.7.c_p
11$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.11.a_h
13$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.13.a_c
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.d_bi
19$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.19.a_ao
23$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.23.c_br
29$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.29.a_ba
31$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.31.g_cp
37$C_2^2$ \( 1 - 28 T^{2} + p^{2} T^{4} \) 2.37.a_abc
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.41.m_de
43$C_2^2$ \( 1 + 29 T^{2} + p^{2} T^{4} \) 2.43.a_bd
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.47.ae_ck
53$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \) 2.53.a_ad
59$C_2^2$ \( 1 + 52 T^{2} + p^{2} T^{4} \) 2.59.a_ca
61$C_2^2$ \( 1 - 93 T^{2} + p^{2} T^{4} \) 2.61.a_adp
67$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.67.a_abe
71$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) 2.71.ad_fm
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.a_eg
79$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 - 12 T + p T^{2} ) \) 2.79.az_mc
83$C_2^2$ \( 1 + 104 T^{2} + p^{2} T^{4} \) 2.83.a_ea
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.89.ag_fi
97$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.97.ac_fq
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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

−10.77531303983255008559116387843, −10.21985587538563896827707054823, −9.592620567118199933389508361059, −9.201153887590441525536293227898, −8.724476398854482766260409322861, −8.132537877364967401947829339637, −7.71501984380188138885296291883, −6.78859390398537507790263659092, −6.31472782023613236968088385918, −5.47462029524128473370343160425, −4.94381681711379579666073686431, −3.91016962134684384975411586577, −3.25116251407748920461240095866, −2.04632942007222918836771120421, 0, 2.04632942007222918836771120421, 3.25116251407748920461240095866, 3.91016962134684384975411586577, 4.94381681711379579666073686431, 5.47462029524128473370343160425, 6.31472782023613236968088385918, 6.78859390398537507790263659092, 7.71501984380188138885296291883, 8.132537877364967401947829339637, 8.724476398854482766260409322861, 9.201153887590441525536293227898, 9.592620567118199933389508361059, 10.21985587538563896827707054823, 10.77531303983255008559116387843

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