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Properties

Label 2-1155-7.4-c1-0-36
Degree $2$
Conductor $1155$
Sign $0.416 + 0.909i$
Analytic cond. $9.22272$
Root an. cond. $3.03689$
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  + (0.307 + 0.532i)2-s + (−0.5 + 0.866i)3-s + (0.811 − 1.40i)4-s + (0.5 + 0.866i)5-s − 0.614·6-s + (−2.24 − 1.39i)7-s + 2.22·8-s + (−0.499 − 0.866i)9-s + (−0.307 + 0.532i)10-s + (0.5 − 0.866i)11-s + (0.811 + 1.40i)12-s − 3.55·13-s + (0.0500 − 1.62i)14-s − 0.999·15-s + (−0.938 − 1.62i)16-s + (2.62 − 4.54i)17-s + ⋯
L(s)  = 1  + (0.217 + 0.376i)2-s + (−0.288 + 0.499i)3-s + (0.405 − 0.702i)4-s + (0.223 + 0.387i)5-s − 0.250·6-s + (−0.850 − 0.526i)7-s + 0.787·8-s + (−0.166 − 0.288i)9-s + (−0.0971 + 0.168i)10-s + (0.150 − 0.261i)11-s + (0.234 + 0.405i)12-s − 0.986·13-s + (0.0133 − 0.434i)14-s − 0.258·15-s + (−0.234 − 0.406i)16-s + (0.636 − 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.416 + 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.416 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.416 + 0.909i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ 0.416 + 0.909i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.335985929\)
\(L(\frac12)\) \(\approx\) \(1.335985929\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (2.24 + 1.39i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-0.307 - 0.532i)T + (-1 + 1.73i)T^{2} \)
13 \( 1 + 3.55T + 13T^{2} \)
17 \( 1 + (-2.62 + 4.54i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.53 + 6.12i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.20 - 3.81i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.14T + 29T^{2} \)
31 \( 1 + (-2.94 + 5.09i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.94 - 3.37i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.75T + 41T^{2} \)
43 \( 1 - 5.22T + 43T^{2} \)
47 \( 1 + (5.57 + 9.65i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.59 + 2.76i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.17 + 3.76i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.99 + 12.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.05 - 1.82i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.22T + 71T^{2} \)
73 \( 1 + (3.26 - 5.66i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.97 - 6.88i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.86T + 83T^{2} \)
89 \( 1 + (2.28 + 3.95i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 11.0T + 97T^{2} \)
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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

−9.775811183348670067649236116968, −9.199127612754120282545743442577, −7.66622344306728803932650141592, −6.89611277085170660367587669669, −6.41079597301443531503947617779, −5.34356042532046143914544819707, −4.72864934127988713030217549383, −3.43531344887610287736243296230, −2.37691245814537561640849331859, −0.53382867132492671013535983930, 1.64019469784967955723715727147, 2.59625924125035741939843672629, 3.64718141730950545227032398376, 4.70113197350032686040344162272, 5.88918059911349130897541867379, 6.52070113664028651818492178973, 7.48746934112978462037747895591, 8.223400755152154412196541723676, 9.076209642154096331827653563852, 10.16899504396176947878929053093

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