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Properties

Label 2-34e2-17.16-c1-0-11
Degree $2$
Conductor $1156$
Sign $0.928 + 0.371i$
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  − 0.765i·3-s + 0.765i·5-s − 1.84i·7-s + 2.41·9-s + 2.93i·11-s − 1.17·13-s + 0.585·15-s + 5.41·19-s − 1.41·21-s + 1.39i·23-s + 4.41·25-s − 4.14i·27-s − 5.99i·29-s + 10.3i·31-s + 2.24·33-s + ⋯
L(s)  = 1  − 0.441i·3-s + 0.342i·5-s − 0.698i·7-s + 0.804·9-s + 0.883i·11-s − 0.324·13-s + 0.151·15-s + 1.24·19-s − 0.308·21-s + 0.291i·23-s + 0.882·25-s − 0.797i·27-s − 1.11i·29-s + 1.85i·31-s + 0.390·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.371i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 + 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $0.928 + 0.371i$
Analytic conductor: \(9.23070\)
Root analytic conductor: \(3.03820\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :1/2),\ 0.928 + 0.371i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.785796337\)
\(L(\frac12)\) \(\approx\) \(1.785796337\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.765iT - 3T^{2} \)
5 \( 1 - 0.765iT - 5T^{2} \)
7 \( 1 + 1.84iT - 7T^{2} \)
11 \( 1 - 2.93iT - 11T^{2} \)
13 \( 1 + 1.17T + 13T^{2} \)
19 \( 1 - 5.41T + 19T^{2} \)
23 \( 1 - 1.39iT - 23T^{2} \)
29 \( 1 + 5.99iT - 29T^{2} \)
31 \( 1 - 10.3iT - 31T^{2} \)
37 \( 1 + 4.90iT - 37T^{2} \)
41 \( 1 + 8.15iT - 41T^{2} \)
43 \( 1 - 4.24T + 43T^{2} \)
47 \( 1 + 7.65T + 47T^{2} \)
53 \( 1 - 13.8T + 53T^{2} \)
59 \( 1 + 5.89T + 59T^{2} \)
61 \( 1 - 0.765iT - 61T^{2} \)
67 \( 1 - 11.3T + 67T^{2} \)
71 \( 1 + 8.15iT - 71T^{2} \)
73 \( 1 - 5.09iT - 73T^{2} \)
79 \( 1 - 9.05iT - 79T^{2} \)
83 \( 1 - 8.24T + 83T^{2} \)
89 \( 1 + 5.17T + 89T^{2} \)
97 \( 1 + 13.3iT - 97T^{2} \)
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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

−9.926230077040438146715751320762, −9.004335888418394142960352497877, −7.80183254389003923751895735679, −7.15468893519581889762326960466, −6.80844564734083088509520773917, −5.45319106651911124719795088446, −4.51689052339115372764117424375, −3.56770241358594830618228152951, −2.27258364727853046504075356891, −1.04936763762669018827171716013, 1.11234077712087685655023986005, 2.64967125008814168515479026833, 3.67369838689585579761413684610, 4.77752996823259839324638920646, 5.44078373208377322203540360162, 6.43488235905142880031978148085, 7.43561926352692143307335722127, 8.314482765684096546035663343581, 9.153426642355683973786112287000, 9.706142876197358970002559373104

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