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Properties

Label 2-34e2-68.47-c0-0-1
Degree $2$
Conductor $1156$
Sign $0.615 + 0.788i$
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  − i·2-s − 4-s + i·8-s + i·9-s + 2·13-s + 16-s + 18-s − i·25-s − 2i·26-s − i·32-s − i·36-s − i·49-s − 50-s − 2·52-s + 2i·53-s + ⋯
L(s)  = 1  − i·2-s − 4-s + i·8-s + i·9-s + 2·13-s + 16-s + 18-s − i·25-s − 2i·26-s − i·32-s − i·36-s − i·49-s − 50-s − 2·52-s + 2i·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $0.615 + 0.788i$
Analytic conductor: \(0.576919\)
Root analytic conductor: \(0.759551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :0),\ 0.615 + 0.788i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.010881572\)
\(L(\frac12)\) \(\approx\) \(1.010881572\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−10.14911886812605344966685252029, −9.042770923072000633582516740499, −8.443073143215756793929554480558, −7.73351118326242519059924752766, −6.34509077012111079513952371251, −5.48307811789978765772994242419, −4.46382693193862481546202965180, −3.63349675825252596942810395588, −2.50896669731885138429607978795, −1.34928060120166883895863755008, 1.20800603005854348550064001621, 3.39118794709560083732820756058, 3.96952724505042743645441889020, 5.21657532887193524305873784777, 6.12474227824019852974080244415, 6.60075638751644783424001245416, 7.61291123898062320768263169986, 8.528243929486229495731371508990, 9.042761878918763567807525692038, 9.828436684303266026564496717583

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