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Properties

Label 2-56e2-64.51-c0-0-0
Degree $2$
Conductor $3136$
Sign $0.634 + 0.773i$
Analytic cond. $1.56506$
Root an. cond. $1.25102$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.923 − 0.382i)2-s + (0.707 − 0.707i)4-s + (0.382 − 0.923i)8-s + (0.382 − 0.923i)9-s + (1.63 + 1.08i)11-s − i·16-s − i·18-s + (1.92 + 0.382i)22-s + (−0.707 + 1.70i)23-s + (−0.923 + 0.382i)25-s + (−1.08 − 1.63i)29-s + (−0.382 − 0.923i)32-s + (−0.382 − 0.923i)36-s + (0.216 + 1.08i)37-s + (0.923 − 1.38i)43-s + (1.92 − 0.382i)44-s + ⋯
L(s)  = 1  + (0.923 − 0.382i)2-s + (0.707 − 0.707i)4-s + (0.382 − 0.923i)8-s + (0.382 − 0.923i)9-s + (1.63 + 1.08i)11-s − i·16-s − i·18-s + (1.92 + 0.382i)22-s + (−0.707 + 1.70i)23-s + (−0.923 + 0.382i)25-s + (−1.08 − 1.63i)29-s + (−0.382 − 0.923i)32-s + (−0.382 − 0.923i)36-s + (0.216 + 1.08i)37-s + (0.923 − 1.38i)43-s + (1.92 − 0.382i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.634 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.634 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3136\)    =    \(2^{6} \cdot 7^{2}\)
Sign: $0.634 + 0.773i$
Analytic conductor: \(1.56506\)
Root analytic conductor: \(1.25102\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3136} (883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3136,\ (\ :0),\ 0.634 + 0.773i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.443252542\)
\(L(\frac12)\) \(\approx\) \(2.443252542\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.923 + 0.382i)T \)
7 \( 1 \)
good3 \( 1 + (-0.382 + 0.923i)T^{2} \)
5 \( 1 + (0.923 - 0.382i)T^{2} \)
11 \( 1 + (-1.63 - 1.08i)T + (0.382 + 0.923i)T^{2} \)
13 \( 1 + (0.923 + 0.382i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + (0.923 + 0.382i)T^{2} \)
23 \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \)
29 \( 1 + (1.08 + 1.63i)T + (-0.382 + 0.923i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-0.216 - 1.08i)T + (-0.923 + 0.382i)T^{2} \)
41 \( 1 + (0.707 + 0.707i)T^{2} \)
43 \( 1 + (-0.923 + 1.38i)T + (-0.382 - 0.923i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (0.923 - 1.38i)T + (-0.382 - 0.923i)T^{2} \)
59 \( 1 + (-0.923 + 0.382i)T^{2} \)
61 \( 1 + (0.382 - 0.923i)T^{2} \)
67 \( 1 + (0.617 + 0.923i)T + (-0.382 + 0.923i)T^{2} \)
71 \( 1 + (-1.30 + 0.541i)T + (0.707 - 0.707i)T^{2} \)
73 \( 1 + (-0.707 - 0.707i)T^{2} \)
79 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
83 \( 1 + (0.923 + 0.382i)T^{2} \)
89 \( 1 + (0.707 - 0.707i)T^{2} \)
97 \( 1 - T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.173076654595093692184925468219, −7.64779482233886695729743657636, −7.19471622864104705088145759997, −6.23847320498916750734585493207, −5.87876909267533405871652446186, −4.66874023687468803077415488403, −3.91533219541736707799007938796, −3.56163649779915565333617582521, −2.07955016107112814486656290034, −1.35756051893200187992218666904, 1.58171695141887322896536789898, 2.60239177171093159873276027463, 3.73875893566002656696733794423, 4.20759869361326278717697421193, 5.15872997497439854253544058159, 6.00103155081729646028966668070, 6.54187463797437879204492854928, 7.33686161366944164651276040040, 8.149785558126710232574722559181, 8.727286031061737077687461692729

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