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Properties

Label 2-34e2-17.8-c1-0-20
Degree $2$
Conductor $1156$
Sign $-0.673 + 0.739i$
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  + (0.707 − 1.70i)3-s + (−1.70 − 0.707i)5-s + (0.707 − 0.292i)7-s + (−0.292 − 0.292i)9-s + (1.29 + 3.12i)11-s − 6.82i·13-s + (−2.41 + 2.41i)15-s + (1.82 − 1.82i)19-s − 1.41i·21-s + (−2.70 − 6.53i)23-s + (−1.12 − 1.12i)25-s + (4.41 − 1.82i)27-s + (−3.70 − 1.53i)29-s + (−2.46 + 5.94i)31-s + 6.24·33-s + ⋯
L(s)  = 1  + (0.408 − 0.985i)3-s + (−0.763 − 0.316i)5-s + (0.267 − 0.110i)7-s + (−0.0976 − 0.0976i)9-s + (0.389 + 0.941i)11-s − 1.89i·13-s + (−0.623 + 0.623i)15-s + (0.419 − 0.419i)19-s − 0.308i·21-s + (−0.564 − 1.36i)23-s + (−0.224 − 0.224i)25-s + (0.849 − 0.351i)27-s + (−0.688 − 0.285i)29-s + (−0.442 + 1.06i)31-s + 1.08·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.673 + 0.739i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.673 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $-0.673 + 0.739i$
Analytic conductor: \(9.23070\)
Root analytic conductor: \(3.03820\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (977, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :1/2),\ -0.673 + 0.739i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.444563120\)
\(L(\frac12)\) \(\approx\) \(1.444563120\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 + 1.70i)T + (-2.12 - 2.12i)T^{2} \)
5 \( 1 + (1.70 + 0.707i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (-0.707 + 0.292i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (-1.29 - 3.12i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + 6.82iT - 13T^{2} \)
19 \( 1 + (-1.82 + 1.82i)T - 19iT^{2} \)
23 \( 1 + (2.70 + 6.53i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (3.70 + 1.53i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (2.46 - 5.94i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (-2.53 + 6.12i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-1.12 + 0.464i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (3 + 3i)T + 43iT^{2} \)
47 \( 1 - 3.65iT - 47T^{2} \)
53 \( 1 + (4.17 - 4.17i)T - 53iT^{2} \)
59 \( 1 + (9.82 + 9.82i)T + 59iT^{2} \)
61 \( 1 + (1.70 - 0.707i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 - 11.3T + 67T^{2} \)
71 \( 1 + (0.464 - 1.12i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-7.94 - 3.29i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (4.36 + 10.5i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (-0.171 + 0.171i)T - 83iT^{2} \)
89 \( 1 - 10.8iT - 89T^{2} \)
97 \( 1 + (0.878 + 0.363i)T + (68.5 + 68.5i)T^{2} \)
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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

−9.384978526293427339238273265672, −8.302081859750382173913757064649, −7.85457885621622311858466562735, −7.28846800774710931406305052457, −6.32385961717832181671257299981, −5.12881793928558148409793122807, −4.25749387665970602815019716802, −3.05962988669955513621383207793, −1.93026343210807458172928602010, −0.60952844784791129359989153613, 1.70749088096903211172726536117, 3.37776589118315397709221371196, 3.82429687822042760935418534070, 4.66012374622699705136160124783, 5.84307441723520210016323901568, 6.83706608667388395700894278009, 7.74058272937132498343002652536, 8.581303988683822672312940146657, 9.444002310747329431688850237500, 9.766411511015618361159590291193

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