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Properties

Label 4-17056-1.1-c1e2-0-0
Degree $4$
Conductor $17056$
Sign $1$
Analytic cond. $1.08750$
Root an. cond. $1.02119$
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 + 6·5-s − 8-s − 2·9-s − 6·10-s − 6·13-s + 16-s + 3·17-s + 2·18-s + 6·20-s + 17·25-s + 6·26-s + 6·29-s − 32-s − 3·34-s − 2·36-s − 14·37-s − 6·40-s + 2·41-s − 12·45-s + 5·49-s − 17·50-s − 6·52-s + 6·53-s − 6·58-s − 8·61-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 2.68·5-s − 0.353·8-s − 2/3·9-s − 1.89·10-s − 1.66·13-s + 1/4·16-s + 0.727·17-s + 0.471·18-s + 1.34·20-s + 17/5·25-s + 1.17·26-s + 1.11·29-s − 0.176·32-s − 0.514·34-s − 1/3·36-s − 2.30·37-s − 0.948·40-s + 0.312·41-s − 1.78·45-s + 5/7·49-s − 2.40·50-s − 0.832·52-s + 0.824·53-s − 0.787·58-s − 1.02·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17056 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(17056\)    =    \(2^{5} \cdot 13 \cdot 41\)
Sign: $1$
Analytic conductor: \(1.08750\)
Root analytic conductor: \(1.02119\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 17056,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.141525018\)
\(L(\frac12)\) \(\approx\) \(1.141525018\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
13$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 7 T + p T^{2} ) \)
41$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 3 T + p T^{2} ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
5$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.5.ag_t
7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.7.a_af
11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.11.a_ac
17$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.17.ad_q
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.19.a_k
23$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.23.a_h
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.29.ag_cg
31$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \) 2.31.a_abg
37$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.o_ek
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.43.a_ao
47$C_2^2$ \( 1 + 31 T^{2} + p^{2} T^{4} \) 2.47.a_bf
53$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.53.ag_db
59$C_2^2$ \( 1 - 29 T^{2} + p^{2} T^{4} \) 2.59.a_abd
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.i_dy
67$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.67.a_ack
71$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.71.a_adf
73$C_2$$\times$$C_2$ \( ( 1 + 13 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.bd_nq
79$C_2^2$ \( 1 - 92 T^{2} + p^{2} T^{4} \) 2.79.a_ado
83$C_2^2$ \( 1 + 55 T^{2} + p^{2} T^{4} \) 2.83.a_cd
89$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.g_gw
97$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.97.ae_hh
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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

−10.55178201275408412467914214539, −10.19681652548814757355408588397, −10.07940402410385058732558201006, −9.496319287642506419535775153376, −8.854044482581721321654852130433, −8.665883377960871450811157052320, −7.54613586547666154045767971386, −7.11222599753414390824770736179, −6.33649247439105383947918831290, −5.80149104039127914052882119657, −5.41596803163749763838624586132, −4.72822548223313873764571061493, −3.06750440219111394092204415468, −2.41518356258620988117739880925, −1.66702351590882034535484175741, 1.66702351590882034535484175741, 2.41518356258620988117739880925, 3.06750440219111394092204415468, 4.72822548223313873764571061493, 5.41596803163749763838624586132, 5.80149104039127914052882119657, 6.33649247439105383947918831290, 7.11222599753414390824770736179, 7.54613586547666154045767971386, 8.665883377960871450811157052320, 8.854044482581721321654852130433, 9.496319287642506419535775153376, 10.07940402410385058732558201006, 10.19681652548814757355408588397, 10.55178201275408412467914214539

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