Properties

Label 2-34e2-68.15-c0-0-2
Degree $2$
Conductor $1156$
Sign $0.362 - 0.931i$
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 + (1.30 − 0.541i)5-s + (−0.707 + 0.707i)8-s + (−0.707 + 0.707i)9-s + (1.30 + 0.541i)10-s − 1.00·16-s − 1.00·18-s + (0.541 + 1.30i)20-s + (0.707 − 0.707i)25-s + (1.30 − 0.541i)29-s + (−0.707 − 0.707i)32-s + (−0.707 − 0.707i)36-s + (−0.541 − 1.30i)37-s + (−0.541 + 1.30i)40-s + (−1.30 − 0.541i)41-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + 1.00i·4-s + (1.30 − 0.541i)5-s + (−0.707 + 0.707i)8-s + (−0.707 + 0.707i)9-s + (1.30 + 0.541i)10-s − 1.00·16-s − 1.00·18-s + (0.541 + 1.30i)20-s + (0.707 − 0.707i)25-s + (1.30 − 0.541i)29-s + (−0.707 − 0.707i)32-s + (−0.707 − 0.707i)36-s + (−0.541 − 1.30i)37-s + (−0.541 + 1.30i)40-s + (−1.30 − 0.541i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.362 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.362 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $0.362 - 0.931i$
Analytic conductor: \(0.576919\)
Root analytic conductor: \(0.759551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (423, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :0),\ 0.362 - 0.931i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.707194583\)
\(L(\frac12)\) \(\approx\) \(1.707194583\)
\(L(1)\) 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 \( 1 + (-1.30 + 0.541i)T + (0.707 - 0.707i)T^{2} \)
7 \( 1 + (-0.707 - 0.707i)T^{2} \)
11 \( 1 + (0.707 + 0.707i)T^{2} \)
13 \( 1 - T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + (0.707 + 0.707i)T^{2} \)
29 \( 1 + (-1.30 + 0.541i)T + (0.707 - 0.707i)T^{2} \)
31 \( 1 + (0.707 - 0.707i)T^{2} \)
37 \( 1 + (0.541 + 1.30i)T + (-0.707 + 0.707i)T^{2} \)
41 \( 1 + (1.30 + 0.541i)T + (0.707 + 0.707i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (1.30 + 0.541i)T + (0.707 + 0.707i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.707 - 0.707i)T^{2} \)
73 \( 1 + (1.30 - 0.541i)T + (0.707 - 0.707i)T^{2} \)
79 \( 1 + (0.707 + 0.707i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (1.30 - 0.541i)T + (0.707 - 0.707i)T^{2} \)
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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

−10.08591467227689503087551101827, −9.067655234085041760830647378905, −8.531666519193108901597068288427, −7.61806593987518484356592469388, −6.58640448322124262982557829053, −5.76981125303707214467429050962, −5.26224424715730027274936228948, −4.37161779233012424439630715427, −2.97284965199853163944970200054, −2.00114764386466823104203924190, 1.49818981190896659507726285148, 2.68459072940010737903354980328, 3.33991680556616519864203997718, 4.70566323157587349048432944680, 5.59975823392301793001808203808, 6.30774194116715655176213671600, 6.87950369622594814077753712615, 8.518216317290326086649521261079, 9.257180468459332813226177020899, 10.13481305311738575195396690354

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