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Properties

Label 2-1096-1096.115-c0-0-0
Degree $2$
Conductor $1096$
Sign $-0.925 + 0.378i$
Analytic cond. $0.546975$
Root an. cond. $0.739577$
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  + (−0.602 + 0.798i)2-s + (0.172 + 1.85i)3-s + (−0.273 − 0.961i)4-s + (−1.58 − 0.981i)6-s + (0.932 + 0.361i)8-s + (−2.43 + 0.455i)9-s + (0.465 + 1.63i)11-s + (1.73 − 0.673i)12-s + (−0.850 + 0.526i)16-s + (−1.58 + 0.614i)17-s + (1.10 − 2.21i)18-s + (0.831 − 1.66i)19-s + (−1.58 − 0.614i)22-s + (−0.510 + 1.79i)24-s + (−0.273 + 0.961i)25-s + ⋯
L(s)  = 1  + (−0.602 + 0.798i)2-s + (0.172 + 1.85i)3-s + (−0.273 − 0.961i)4-s + (−1.58 − 0.981i)6-s + (0.932 + 0.361i)8-s + (−2.43 + 0.455i)9-s + (0.465 + 1.63i)11-s + (1.73 − 0.673i)12-s + (−0.850 + 0.526i)16-s + (−1.58 + 0.614i)17-s + (1.10 − 2.21i)18-s + (0.831 − 1.66i)19-s + (−1.58 − 0.614i)22-s + (−0.510 + 1.79i)24-s + (−0.273 + 0.961i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 + 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 + 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1096\)    =    \(2^{3} \cdot 137\)
Sign: $-0.925 + 0.378i$
Analytic conductor: \(0.546975\)
Root analytic conductor: \(0.739577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1096} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1096,\ (\ :0),\ -0.925 + 0.378i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6688464635\)
\(L(\frac12)\) \(\approx\) \(0.6688464635\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.602 - 0.798i)T \)
137 \( 1 + (-0.445 - 0.895i)T \)
good3 \( 1 + (-0.172 - 1.85i)T + (-0.982 + 0.183i)T^{2} \)
5 \( 1 + (0.273 - 0.961i)T^{2} \)
7 \( 1 + (-0.0922 + 0.995i)T^{2} \)
11 \( 1 + (-0.465 - 1.63i)T + (-0.850 + 0.526i)T^{2} \)
13 \( 1 + (-0.0922 + 0.995i)T^{2} \)
17 \( 1 + (1.58 - 0.614i)T + (0.739 - 0.673i)T^{2} \)
19 \( 1 + (-0.831 + 1.66i)T + (-0.602 - 0.798i)T^{2} \)
23 \( 1 + (-0.445 + 0.895i)T^{2} \)
29 \( 1 + (-0.445 + 0.895i)T^{2} \)
31 \( 1 + (0.602 - 0.798i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 0.184T + T^{2} \)
43 \( 1 + (0.757 - 1.52i)T + (-0.602 - 0.798i)T^{2} \)
47 \( 1 + (-0.932 + 0.361i)T^{2} \)
53 \( 1 + (0.602 + 0.798i)T^{2} \)
59 \( 1 + (-1.18 + 0.221i)T + (0.932 - 0.361i)T^{2} \)
61 \( 1 + (-0.932 - 0.361i)T^{2} \)
67 \( 1 + (-1.37 + 1.25i)T + (0.0922 - 0.995i)T^{2} \)
71 \( 1 + (0.850 + 0.526i)T^{2} \)
73 \( 1 + (-0.136 - 0.124i)T + (0.0922 + 0.995i)T^{2} \)
79 \( 1 + (0.982 + 0.183i)T^{2} \)
83 \( 1 + (-1.37 - 0.533i)T + (0.739 + 0.673i)T^{2} \)
89 \( 1 + (0.537 + 0.711i)T + (-0.273 + 0.961i)T^{2} \)
97 \( 1 + (-0.149 - 0.526i)T + (-0.850 + 0.526i)T^{2} \)
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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

−10.15767087059313485982452918702, −9.492901236601751550358796216931, −9.179438609824371975681273870048, −8.336967047780307265669978794898, −7.19869116722620540753695516656, −6.39189317493975448903603153267, −5.04652240100537501613148625343, −4.76145714019397996971401239360, −3.80736754372184021537044635554, −2.24885912363550974204577617232, 0.73137256922889119150712694412, 1.88774110119263994760970423241, 2.84263900123823060117526063964, 3.83264539119007287827956391225, 5.60850151790006321229554102839, 6.50188828864163367825267580126, 7.23359093606245978119823547129, 8.229581750036554284688808853589, 8.493412029419153263297438646353, 9.387818597478701830976200125892

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