Properties

Label 2-1155-33.32-c1-0-62
Degree $2$
Conductor $1155$
Sign $0.557 + 0.830i$
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  − 0.490·2-s + (1.73 + 0.0687i)3-s − 1.75·4-s i·5-s + (−0.849 − 0.0337i)6-s i·7-s + 1.84·8-s + (2.99 + 0.237i)9-s + 0.490i·10-s + (1.95 + 2.67i)11-s + (−3.04 − 0.120i)12-s − 2.97i·13-s + 0.490i·14-s + (0.0687 − 1.73i)15-s + 2.61·16-s − 3.37·17-s + ⋯
L(s)  = 1  − 0.346·2-s + (0.999 + 0.0396i)3-s − 0.879·4-s − 0.447i·5-s + (−0.346 − 0.0137i)6-s − 0.377i·7-s + 0.652·8-s + (0.996 + 0.0792i)9-s + 0.155i·10-s + (0.590 + 0.807i)11-s + (−0.878 − 0.0348i)12-s − 0.826i·13-s + 0.131i·14-s + (0.0177 − 0.446i)15-s + 0.653·16-s − 0.817·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 + 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.557 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.557 + 0.830i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (1121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ 0.557 + 0.830i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.585498200\)
\(L(\frac12)\) \(\approx\) \(1.585498200\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.73 - 0.0687i)T \)
5 \( 1 + iT \)
7 \( 1 + iT \)
11 \( 1 + (-1.95 - 2.67i)T \)
good2 \( 1 + 0.490T + 2T^{2} \)
13 \( 1 + 2.97iT - 13T^{2} \)
17 \( 1 + 3.37T + 17T^{2} \)
19 \( 1 + 5.83iT - 19T^{2} \)
23 \( 1 - 0.534iT - 23T^{2} \)
29 \( 1 + 4.74T + 29T^{2} \)
31 \( 1 - 7.76T + 31T^{2} \)
37 \( 1 - 1.78T + 37T^{2} \)
41 \( 1 - 8.07T + 41T^{2} \)
43 \( 1 + 2.55iT - 43T^{2} \)
47 \( 1 + 7.04iT - 47T^{2} \)
53 \( 1 - 3.11iT - 53T^{2} \)
59 \( 1 + 5.20iT - 59T^{2} \)
61 \( 1 + 9.92iT - 61T^{2} \)
67 \( 1 - 2.50T + 67T^{2} \)
71 \( 1 + 11.7iT - 71T^{2} \)
73 \( 1 + 5.53iT - 73T^{2} \)
79 \( 1 + 1.72iT - 79T^{2} \)
83 \( 1 - 16.9T + 83T^{2} \)
89 \( 1 - 6.67iT - 89T^{2} \)
97 \( 1 - 3.15T + 97T^{2} \)
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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

−9.352687361081237531145907907231, −9.063173520755928942512866438482, −8.114178252275245149069575035354, −7.52344403718382982346867241363, −6.55779268820057025025754819248, −5.00648410551108766022388853250, −4.43717194605873150594268756049, −3.55232536580310628913152086187, −2.18574485317149265993435856295, −0.802984907739571913911406156453, 1.37080689240699631717408177795, 2.65335478603700146063302142405, 3.83984795512311538627467588290, 4.37287364498543713401613062335, 5.80127827813270172048784517737, 6.71022364896087714854489360282, 7.77658316710411105536509199900, 8.395226371494594692880097012489, 9.115368844518877650443825261924, 9.611661698898147197030827771153

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