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Properties

Label 2-34e2-17.2-c1-0-2
Degree $2$
Conductor $1156$
Sign $-0.275 - 0.961i$
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  + (−2.52 + 1.04i)3-s + (−1.32 − 3.20i)5-s + (−1.04 + 2.52i)7-s + (3.15 − 3.15i)9-s + (−1.17 − 0.485i)11-s − 5.46i·13-s + (6.69 + 6.69i)15-s + (−1.03 − 1.03i)19-s − 7.46i·21-s + (1.17 + 0.485i)23-s + (−4.94 + 4.94i)25-s + (−1.53 + 3.69i)27-s + (−1.32 − 3.20i)29-s + (3.87 − 1.60i)31-s + 3.46·33-s + ⋯
L(s)  = 1  + (−1.45 + 0.603i)3-s + (−0.592 − 1.43i)5-s + (−0.395 + 0.954i)7-s + (1.05 − 1.05i)9-s + (−0.353 − 0.146i)11-s − 1.51i·13-s + (1.72 + 1.72i)15-s + (−0.237 − 0.237i)19-s − 1.62i·21-s + (0.244 + 0.101i)23-s + (−0.989 + 0.989i)25-s + (−0.294 + 0.711i)27-s + (−0.246 − 0.594i)29-s + (0.696 − 0.288i)31-s + 0.603·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.275 - 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.275 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1156\)    =    \(2^{2} \cdot 17^{2}\)
Sign: $-0.275 - 0.961i$
Analytic conductor: \(9.23070\)
Root analytic conductor: \(3.03820\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1156} (733, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1156,\ (\ :1/2),\ -0.275 - 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2672395393\)
\(L(\frac12)\) \(\approx\) \(0.2672395393\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.52 - 1.04i)T + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (1.32 + 3.20i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (1.04 - 2.52i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (1.17 + 0.485i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + 5.46iT - 13T^{2} \)
19 \( 1 + (1.03 + 1.03i)T + 19iT^{2} \)
23 \( 1 + (-1.17 - 0.485i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (1.32 + 3.20i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (-3.87 + 1.60i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (4.19 - 1.73i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (2.29 - 5.54i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (5.93 - 5.93i)T - 43iT^{2} \)
47 \( 1 - 6.92iT - 47T^{2} \)
53 \( 1 + (-9.14 - 9.14i)T + 53iT^{2} \)
59 \( 1 + (1.79 - 1.79i)T - 59iT^{2} \)
61 \( 1 + (-0.205 + 0.495i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + 14.9T + 67T^{2} \)
71 \( 1 + (-7.57 + 3.13i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (-0.765 - 1.84i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (11.2 + 4.66i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (-1.79 - 1.79i)T + 83iT^{2} \)
89 \( 1 - 2.53iT - 89T^{2} \)
97 \( 1 + (-1.88 - 4.55i)T + (-68.5 + 68.5i)T^{2} \)
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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

−10.11311443731527758316555721972, −9.285526589838840612835757235265, −8.461662361208195500944606006130, −7.72337889553738157890754393879, −6.26755024870193871321842341805, −5.63305292599271496620715196304, −5.01946548934054809249113293756, −4.32949864192438787625550620966, −2.97701346031119652947654673266, −0.935039929673078303302323424031, 0.18491438191775380294537965242, 1.89109072536710963788794387088, 3.41893587666709115152548282594, 4.31526352226140239796940678863, 5.45513487569375410357809238698, 6.56209771394876036212856362122, 6.98813373802779280243248753618, 7.26781835872995146678415975705, 8.616760478786206135674877193859, 10.08037204787091706858140123555

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