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Properties

Label 2-2310-33.32-c1-0-87
Degree $2$
Conductor $2310$
Sign $-0.573 + 0.819i$
Analytic cond. $18.4454$
Root an. cond. $4.29481$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + (−0.292 + 1.70i)3-s + 4-s + i·5-s + (−0.292 + 1.70i)6-s + i·7-s + 8-s + (−2.82 − i)9-s + i·10-s + (−3 − 1.41i)11-s + (−0.292 + 1.70i)12-s − 6.24i·13-s + i·14-s + (−1.70 − 0.292i)15-s + 16-s − 4.58·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.169 + 0.985i)3-s + 0.5·4-s + 0.447i·5-s + (−0.119 + 0.696i)6-s + 0.377i·7-s + 0.353·8-s + (−0.942 − 0.333i)9-s + 0.316i·10-s + (−0.904 − 0.426i)11-s + (−0.0845 + 0.492i)12-s − 1.73i·13-s + 0.267i·14-s + (−0.440 − 0.0756i)15-s + 0.250·16-s − 1.11·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.573 + 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.573 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2310\)    =    \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.573 + 0.819i$
Analytic conductor: \(18.4454\)
Root analytic conductor: \(4.29481\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2310} (1121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 2310,\ (\ :1/2),\ -0.573 + 0.819i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (0.292 - 1.70i)T \)
5 \( 1 - iT \)
7 \( 1 - iT \)
11 \( 1 + (3 + 1.41i)T \)
good13 \( 1 + 6.24iT - 13T^{2} \)
17 \( 1 + 4.58T + 17T^{2} \)
19 \( 1 - 2.82iT - 19T^{2} \)
23 \( 1 - 0.828iT - 23T^{2} \)
29 \( 1 + 6.82T + 29T^{2} \)
31 \( 1 + 5.41T + 31T^{2} \)
37 \( 1 + 2.82T + 37T^{2} \)
41 \( 1 + 11.0T + 41T^{2} \)
43 \( 1 + 0.828iT - 43T^{2} \)
47 \( 1 + 0.585iT - 47T^{2} \)
53 \( 1 - 13.3iT - 53T^{2} \)
59 \( 1 + 13.8iT - 59T^{2} \)
61 \( 1 + 2.58iT - 61T^{2} \)
67 \( 1 - 13.3T + 67T^{2} \)
71 \( 1 - 11.3iT - 71T^{2} \)
73 \( 1 - 4.58iT - 73T^{2} \)
79 \( 1 + 10iT - 79T^{2} \)
83 \( 1 + 16T + 83T^{2} \)
89 \( 1 - 7.65iT - 89T^{2} \)
97 \( 1 - 3.65T + 97T^{2} \)
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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

−8.657515242019352558362102980801, −8.040618002222345906319721519438, −7.09955480617268828036045543243, −5.96918519113663887818558536533, −5.52808118834120429760738682833, −4.88709933198078772139326126983, −3.68183406984193744591738015550, −3.18277547183074997119217515298, −2.21886442683209112316902075158, 0, 1.73901622704912851425749804825, 2.27693943019107075826401901297, 3.63439173177617295695501921936, 4.64248462756593832058402128145, 5.22361876697336580069539705825, 6.24442177772809096784932775531, 7.02635314711906471486151563408, 7.32652570770830049208197074936, 8.465513645496616830223372761681

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