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Properties

Label 2-34e2-17.15-c1-0-17
Degree $2$
Conductor $1156$
Sign $-0.440 + 0.897i$
Analytic cond. $9.23070$
Root an. cond. $3.03820$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.04 − 2.52i)3-s + (3.20 − 1.32i)5-s + (2.52 + 1.04i)7-s + (−3.15 + 3.15i)9-s + (−0.485 + 1.17i)11-s − 5.46i·13-s + (−6.69 − 6.69i)15-s + (1.03 + 1.03i)19-s − 7.46i·21-s + (0.485 − 1.17i)23-s + (4.94 − 4.94i)25-s + (3.69 + 1.53i)27-s + (3.20 − 1.32i)29-s + (1.60 + 3.87i)31-s + 3.46·33-s + ⋯
L(s)  = 1  + (−0.603 − 1.45i)3-s + (1.43 − 0.592i)5-s + (0.954 + 0.395i)7-s + (−1.05 + 1.05i)9-s + (−0.146 + 0.353i)11-s − 1.51i·13-s + (−1.72 − 1.72i)15-s + (0.237 + 0.237i)19-s − 1.62i·21-s + (0.101 − 0.244i)23-s + (0.989 − 0.989i)25-s + (0.711 + 0.294i)27-s + (0.594 − 0.246i)29-s + (0.288 + 0.696i)31-s + 0.603·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.440 + 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.440 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $-0.440 + 0.897i$
Analytic conductor: \(9.23070\)
Root analytic conductor: \(3.03820\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (1001, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :1/2),\ -0.440 + 0.897i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.859168934\)
\(L(\frac12)\) \(\approx\) \(1.859168934\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 \)
good3 \( 1 + (1.04 + 2.52i)T + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (-3.20 + 1.32i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (-2.52 - 1.04i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (0.485 - 1.17i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + 5.46iT - 13T^{2} \)
19 \( 1 + (-1.03 - 1.03i)T + 19iT^{2} \)
23 \( 1 + (-0.485 + 1.17i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-3.20 + 1.32i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (-1.60 - 3.87i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (1.73 + 4.19i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-5.54 - 2.29i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (-5.93 + 5.93i)T - 43iT^{2} \)
47 \( 1 - 6.92iT - 47T^{2} \)
53 \( 1 + (9.14 + 9.14i)T + 53iT^{2} \)
59 \( 1 + (-1.79 + 1.79i)T - 59iT^{2} \)
61 \( 1 + (0.495 + 0.205i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + 14.9T + 67T^{2} \)
71 \( 1 + (-3.13 - 7.57i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (1.84 - 0.765i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (4.66 - 11.2i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (1.79 + 1.79i)T + 83iT^{2} \)
89 \( 1 - 2.53iT - 89T^{2} \)
97 \( 1 + (4.55 - 1.88i)T + (68.5 - 68.5i)T^{2} \)
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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

−9.527961199673857930977378611471, −8.464963844483122682497752556269, −7.910329526545460718555939875075, −6.98838586990238219024618510903, −5.94726684535822007641386895410, −5.56461948161037331412260810221, −4.79522872437163545627745514690, −2.68880577682459751818474633165, −1.80015164236222983807101234985, −0.944157279257218730697978226314, 1.63016908822917745644037540296, 2.92618517538055371023445150878, 4.26677064315217056992020591063, 4.81852947683034267060759283855, 5.78234921529132805169475901545, 6.39593979076085186924855907855, 7.50031073203040394725270889954, 8.853268439384932022121752219891, 9.435254237756312620330390831280, 10.08906242687581355771527803997

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