Properties

Label 2-34e2-1156.747-c0-0-0
Degree $2$
Conductor $1156$
Sign $-0.887 + 0.460i$
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  + (0.850 − 0.526i)2-s + (0.445 − 0.895i)4-s + (−1.98 − 0.183i)5-s + (−0.0922 − 0.995i)8-s + (−0.932 + 0.361i)9-s + (−1.78 + 0.887i)10-s + (−0.172 − 1.85i)13-s + (−0.602 − 0.798i)16-s + (−0.602 − 0.798i)17-s + (−0.602 + 0.798i)18-s + (−1.04 + 1.69i)20-s + (2.91 + 0.544i)25-s + (−1.12 − 1.48i)26-s + (−0.646 − 0.322i)29-s + (−0.932 − 0.361i)32-s + ⋯
L(s)  = 1  + (0.850 − 0.526i)2-s + (0.445 − 0.895i)4-s + (−1.98 − 0.183i)5-s + (−0.0922 − 0.995i)8-s + (−0.932 + 0.361i)9-s + (−1.78 + 0.887i)10-s + (−0.172 − 1.85i)13-s + (−0.602 − 0.798i)16-s + (−0.602 − 0.798i)17-s + (−0.602 + 0.798i)18-s + (−1.04 + 1.69i)20-s + (2.91 + 0.544i)25-s + (−1.12 − 1.48i)26-s + (−0.646 − 0.322i)29-s + (−0.932 − 0.361i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $-0.887 + 0.460i$
Analytic conductor: \(0.576919\)
Root analytic conductor: \(0.759551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (747, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :0),\ -0.887 + 0.460i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8378846596\)
\(L(\frac12)\) \(\approx\) \(0.8378846596\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.850 + 0.526i)T \)
17 \( 1 + (0.602 + 0.798i)T \)
good3 \( 1 + (0.932 - 0.361i)T^{2} \)
5 \( 1 + (1.98 + 0.183i)T + (0.982 + 0.183i)T^{2} \)
7 \( 1 + (0.739 - 0.673i)T^{2} \)
11 \( 1 + (-0.602 - 0.798i)T^{2} \)
13 \( 1 + (0.172 + 1.85i)T + (-0.982 + 0.183i)T^{2} \)
19 \( 1 + (-0.445 - 0.895i)T^{2} \)
23 \( 1 + (0.739 - 0.673i)T^{2} \)
29 \( 1 + (0.646 + 0.322i)T + (0.602 + 0.798i)T^{2} \)
31 \( 1 + (-0.982 + 0.183i)T^{2} \)
37 \( 1 + (-1.01 + 0.288i)T + (0.850 - 0.526i)T^{2} \)
41 \( 1 + (-0.365 - 1.95i)T + (-0.932 + 0.361i)T^{2} \)
43 \( 1 + (0.273 + 0.961i)T^{2} \)
47 \( 1 + (-0.739 - 0.673i)T^{2} \)
53 \( 1 + (-1.37 + 0.533i)T + (0.739 - 0.673i)T^{2} \)
59 \( 1 + (-0.0922 - 0.995i)T^{2} \)
61 \( 1 + (0.247 + 0.271i)T + (-0.0922 + 0.995i)T^{2} \)
67 \( 1 + (-0.445 - 0.895i)T^{2} \)
71 \( 1 + (0.739 - 0.673i)T^{2} \)
73 \( 1 + (-0.576 + 0.435i)T + (0.273 - 0.961i)T^{2} \)
79 \( 1 + (0.445 + 0.895i)T^{2} \)
83 \( 1 + (-0.932 - 0.361i)T^{2} \)
89 \( 1 + (-0.156 + 1.69i)T + (-0.982 - 0.183i)T^{2} \)
97 \( 1 + (0.646 + 1.66i)T + (-0.739 + 0.673i)T^{2} \)
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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

−9.856489586531812806575103345194, −8.683405276725889810948310710926, −7.88210896094076673316739811203, −7.32373339080767102461169709476, −6.05458712102985994661277470431, −5.08071819873882782045442033978, −4.41177985844225435506453738981, −3.31560958462054778070525265224, −2.77201516152396707637811786059, −0.54864881468592047112449689459, 2.46460298854711158918327676560, 3.77820371680408745594077474261, 4.02314947690434786689823932411, 5.06457843295597270247734119611, 6.33739812708424034119789978613, 6.97125656628177057379997644457, 7.70661562925226833259022172991, 8.569493632648885459607317843118, 9.045390343664980299117188190740, 10.86703626645144183142887297186

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