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Properties

Label 2-3510-9.7-c1-0-33
Degree $2$
Conductor $3510$
Sign $-0.252 + 0.967i$
Analytic cond. $28.0274$
Root an. cond. $5.29409$
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  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.425 + 0.737i)7-s − 0.999·8-s − 0.999·10-s + (−0.0743 + 0.128i)11-s + (0.5 + 0.866i)13-s + (0.425 + 0.737i)14-s + (−0.5 + 0.866i)16-s + 0.0580·17-s + 1.85·19-s + (−0.499 + 0.866i)20-s + (0.0743 + 0.128i)22-s + (2.78 + 4.82i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.223 − 0.387i)5-s + (−0.160 + 0.278i)7-s − 0.353·8-s − 0.316·10-s + (−0.0224 + 0.0388i)11-s + (0.138 + 0.240i)13-s + (0.113 + 0.197i)14-s + (−0.125 + 0.216i)16-s + 0.0140·17-s + 0.424·19-s + (−0.111 + 0.193i)20-s + (0.0158 + 0.0274i)22-s + (0.581 + 1.00i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3510 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3510 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3510\)    =    \(2 \cdot 3^{3} \cdot 5 \cdot 13\)
Sign: $-0.252 + 0.967i$
Analytic conductor: \(28.0274\)
Root analytic conductor: \(5.29409\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3510} (2341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3510,\ (\ :1/2),\ -0.252 + 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.858960227\)
\(L(\frac12)\) \(\approx\) \(1.858960227\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
5 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
good7 \( 1 + (0.425 - 0.737i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.0743 - 0.128i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 0.0580T + 17T^{2} \)
19 \( 1 - 1.85T + 19T^{2} \)
23 \( 1 + (-2.78 - 4.82i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.26 + 5.66i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.59 + 2.76i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.73T + 37T^{2} \)
41 \( 1 + (4.31 + 7.47i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.27 + 2.21i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.55 + 11.3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 11.8T + 53T^{2} \)
59 \( 1 + (3.17 + 5.49i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.10 + 1.90i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.73 - 4.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.8T + 71T^{2} \)
73 \( 1 + 6.14T + 73T^{2} \)
79 \( 1 + (6.67 - 11.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.96 + 12.0i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 5.88T + 89T^{2} \)
97 \( 1 + (-1.31 + 2.27i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.601919124353669375657277405199, −7.59095230915663462763227454197, −6.91023596006247193998860398382, −5.79679713979438612922982916394, −5.37257322327256040983620583735, −4.35153131268544286277590470802, −3.71514744553861201362336842476, −2.74958044655538897781705317870, −1.79212890742528835742441172445, −0.59295338382071714908947998957, 1.02940111549011006394306798956, 2.65165387262517947677295813795, 3.33036206625799722963860331873, 4.27609877110588659286315397351, 4.99675065709465299815501583346, 5.87585256488379877668883363468, 6.63397924304275654930500162706, 7.20383783397046605469523965413, 7.904898396658160350967040320032, 8.691002668015483656758437201611

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