Properties

Label 2-34e2-1156.599-c0-0-0
Degree $2$
Conductor $1156$
Sign $0.947 - 0.320i$
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  + (0.995 − 0.0922i)2-s + (0.982 − 0.183i)4-s + (0.850 + 1.52i)5-s + (0.961 − 0.273i)8-s + (−0.895 − 0.445i)9-s + (0.987 + 1.44i)10-s + (−0.489 − 1.72i)13-s + (0.932 − 0.361i)16-s + (−0.932 + 0.361i)17-s + (−0.932 − 0.361i)18-s + (1.11 + 1.34i)20-s + (−1.08 + 1.74i)25-s + (−0.646 − 1.66i)26-s + (−1.07 + 1.56i)29-s + (0.895 − 0.445i)32-s + ⋯
L(s)  = 1  + (0.995 − 0.0922i)2-s + (0.982 − 0.183i)4-s + (0.850 + 1.52i)5-s + (0.961 − 0.273i)8-s + (−0.895 − 0.445i)9-s + (0.987 + 1.44i)10-s + (−0.489 − 1.72i)13-s + (0.932 − 0.361i)16-s + (−0.932 + 0.361i)17-s + (−0.932 − 0.361i)18-s + (1.11 + 1.34i)20-s + (−1.08 + 1.74i)25-s + (−0.646 − 1.66i)26-s + (−1.07 + 1.56i)29-s + (0.895 − 0.445i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $0.947 - 0.320i$
Analytic conductor: \(0.576919\)
Root analytic conductor: \(0.759551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (599, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :0),\ 0.947 - 0.320i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.025260282\)
\(L(\frac12)\) \(\approx\) \(2.025260282\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.995 + 0.0922i)T \)
17 \( 1 + (0.932 - 0.361i)T \)
good3 \( 1 + (0.895 + 0.445i)T^{2} \)
5 \( 1 + (-0.850 - 1.52i)T + (-0.526 + 0.850i)T^{2} \)
7 \( 1 + (-0.798 + 0.602i)T^{2} \)
11 \( 1 + (-0.361 - 0.932i)T^{2} \)
13 \( 1 + (0.489 + 1.72i)T + (-0.850 + 0.526i)T^{2} \)
19 \( 1 + (-0.982 - 0.183i)T^{2} \)
23 \( 1 + (-0.798 + 0.602i)T^{2} \)
29 \( 1 + (1.07 - 1.56i)T + (-0.361 - 0.932i)T^{2} \)
31 \( 1 + (-0.526 - 0.850i)T^{2} \)
37 \( 1 + (-1.27 + 0.0590i)T + (0.995 - 0.0922i)T^{2} \)
41 \( 1 + (0.947 + 0.222i)T + (0.895 + 0.445i)T^{2} \)
43 \( 1 + (0.739 + 0.673i)T^{2} \)
47 \( 1 + (0.602 - 0.798i)T^{2} \)
53 \( 1 + (1.42 + 0.711i)T + (0.602 + 0.798i)T^{2} \)
59 \( 1 + (-0.273 - 0.961i)T^{2} \)
61 \( 1 + (-0.268 + 1.92i)T + (-0.961 - 0.273i)T^{2} \)
67 \( 1 + (0.982 + 0.183i)T^{2} \)
71 \( 1 + (0.798 - 0.602i)T^{2} \)
73 \( 1 + (0.256 - 0.581i)T + (-0.673 - 0.739i)T^{2} \)
79 \( 1 + (-0.183 + 0.982i)T^{2} \)
83 \( 1 + (0.445 + 0.895i)T^{2} \)
89 \( 1 + (0.544 - 1.91i)T + (-0.850 - 0.526i)T^{2} \)
97 \( 1 + (0.524 + 1.56i)T + (-0.798 + 0.602i)T^{2} \)
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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

−10.30069927467995364663396003172, −9.517184834404302228360389654043, −8.202175904671347451888333157077, −7.21246947744655740180898756779, −6.51495877804544461148032118156, −5.79877159384346335065276568482, −5.14907783594647636121208698373, −3.53677670260518655986044809428, −2.97885818212187197857339958169, −2.10460164519940506045807827662, 1.77107474032880831894118929692, 2.53223799197842162879504569454, 4.30342108250513758173591076686, 4.66284769051343612368848308010, 5.67084728579025526298956271761, 6.23017182857803668437695885033, 7.35496582056929114683320473129, 8.368489671352762148932275120096, 9.146072860032920200479919877885, 9.776439517479769880224974162322

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