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Properties

Label 2-34e2-68.59-c0-0-0
Degree $2$
Conductor $1156$
Sign $0.739 + 0.673i$
Analytic cond. $0.576919$
Root an. cond. $0.759551$
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  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (0.707 + 0.707i)8-s + (−0.707 − 0.707i)9-s − 2i·13-s − 1.00·16-s + 1.00·18-s + (−0.707 − 0.707i)25-s + (1.41 + 1.41i)26-s + (0.707 − 0.707i)32-s + (−0.707 + 0.707i)36-s + (0.707 − 0.707i)49-s + 1.00·50-s − 2.00·52-s + (1.41 − 1.41i)53-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (0.707 + 0.707i)8-s + (−0.707 − 0.707i)9-s − 2i·13-s − 1.00·16-s + 1.00·18-s + (−0.707 − 0.707i)25-s + (1.41 + 1.41i)26-s + (0.707 − 0.707i)32-s + (−0.707 + 0.707i)36-s + (0.707 − 0.707i)49-s + 1.00·50-s − 2.00·52-s + (1.41 − 1.41i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $0.739 + 0.673i$
Analytic conductor: \(0.576919\)
Root analytic conductor: \(0.759551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (399, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :0),\ 0.739 + 0.673i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6047649598\)
\(L(\frac12)\) \(\approx\) \(0.6047649598\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 \)
good3 \( 1 + (0.707 + 0.707i)T^{2} \)
5 \( 1 + (0.707 + 0.707i)T^{2} \)
7 \( 1 + (-0.707 + 0.707i)T^{2} \)
11 \( 1 + (0.707 - 0.707i)T^{2} \)
13 \( 1 + 2iT - T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 + (0.707 - 0.707i)T^{2} \)
29 \( 1 + (0.707 + 0.707i)T^{2} \)
31 \( 1 + (0.707 + 0.707i)T^{2} \)
37 \( 1 + (-0.707 - 0.707i)T^{2} \)
41 \( 1 + (0.707 - 0.707i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 + (0.707 - 0.707i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.707 + 0.707i)T^{2} \)
73 \( 1 + (0.707 + 0.707i)T^{2} \)
79 \( 1 + (0.707 - 0.707i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - 2iT - T^{2} \)
97 \( 1 + (0.707 + 0.707i)T^{2} \)
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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

−9.915569738276304808931199629022, −8.889634689569386908436472950085, −8.265446210646297376629930137086, −7.59430638131393476972158937878, −6.56293357944531347313049841148, −5.77285885984131962506888804467, −5.16635277011696563143523971531, −3.69877674528110823115230203257, −2.47777126484619914499070930457, −0.68720107346924350493028983890, 1.68419321045449156319243582591, 2.59063033828554983524913583898, 3.84080170621645936810794188847, 4.68533095736488509834066282519, 5.96628239236071102952080350543, 7.05791250135610464860817171784, 7.72128157418238508616334322503, 8.783512828484458317585369240634, 9.145525200130693397355686146848, 10.09035560041178367151190129800

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