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Properties

Label 2-34e2-68.43-c0-0-1
Degree $2$
Conductor $1156$
Sign $0.996 + 0.0883i$
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 + (0.541 − 1.30i)5-s + (0.707 + 0.707i)8-s + (0.707 + 0.707i)9-s + (0.541 + 1.30i)10-s − 1.00·16-s − 1.00·18-s + (−1.30 − 0.541i)20-s + (−0.707 − 0.707i)25-s + (0.541 − 1.30i)29-s + (0.707 − 0.707i)32-s + (0.707 − 0.707i)36-s + (1.30 + 0.541i)37-s + (1.30 − 0.541i)40-s + (−0.541 − 1.30i)41-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (0.541 − 1.30i)5-s + (0.707 + 0.707i)8-s + (0.707 + 0.707i)9-s + (0.541 + 1.30i)10-s − 1.00·16-s − 1.00·18-s + (−1.30 − 0.541i)20-s + (−0.707 − 0.707i)25-s + (0.541 − 1.30i)29-s + (0.707 − 0.707i)32-s + (0.707 − 0.707i)36-s + (1.30 + 0.541i)37-s + (1.30 − 0.541i)40-s + (−0.541 − 1.30i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $0.996 + 0.0883i$
Analytic conductor: \(0.576919\)
Root analytic conductor: \(0.759551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :0),\ 0.996 + 0.0883i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8538643172\)
\(L(\frac12)\) \(\approx\) \(0.8538643172\)
\(L(1)\) 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 \( 1 + (-0.541 + 1.30i)T + (-0.707 - 0.707i)T^{2} \)
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 \( 1 + (-0.541 + 1.30i)T + (-0.707 - 0.707i)T^{2} \)
31 \( 1 + (-0.707 - 0.707i)T^{2} \)
37 \( 1 + (-1.30 - 0.541i)T + (0.707 + 0.707i)T^{2} \)
41 \( 1 + (0.541 + 1.30i)T + (-0.707 + 0.707i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 + (0.541 + 1.30i)T + (-0.707 + 0.707i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.707 - 0.707i)T^{2} \)
73 \( 1 + (0.541 - 1.30i)T + (-0.707 - 0.707i)T^{2} \)
79 \( 1 + (-0.707 + 0.707i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.541 - 1.30i)T + (-0.707 - 0.707i)T^{2} \)
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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

−9.747417002860782660353533437950, −9.179580490617498194907714013210, −8.269546653592853486649756943215, −7.76622313380481874421052616847, −6.68835856260288842742900179958, −5.78746977278885680713039795264, −4.98029479261940137385868674118, −4.32105891984986622507252050702, −2.21668949643421053592745939096, −1.15265089224734115874199161799, 1.49805676899469575772130237740, 2.72851795655925785183081877707, 3.45267243443843036676194193299, 4.56970689425322351384318821212, 6.12689617517964654269192195043, 6.85305114089806502391583147989, 7.47098414072593692703949069015, 8.547434141324302735330289630562, 9.479860754133207028675864116447, 10.01594919405396499173894548305

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