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Properties

Label 2-30-15.8-c9-0-10
Degree $2$
Conductor $30$
Sign $0.874 - 0.485i$
Analytic cond. $15.4510$
Root an. cond. $3.93078$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (11.3 + 11.3i)2-s + (−111. + 85.7i)3-s + 256. i·4-s + (520. − 1.29e3i)5-s + (−2.22e3 − 286. i)6-s + (5.78e3 − 5.78e3i)7-s + (−2.89e3 + 2.89e3i)8-s + (4.98e3 − 1.90e4i)9-s + (2.05e4 − 8.77e3i)10-s + 3.28e4i·11-s + (−2.19e4 − 2.84e4i)12-s + (6.18e4 + 6.18e4i)13-s + 1.30e5·14-s + (5.33e4 + 1.88e5i)15-s − 6.55e4·16-s + (3.26e5 + 3.26e5i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.791 + 0.611i)3-s + 0.500i·4-s + (0.372 − 0.927i)5-s + (−0.701 − 0.0902i)6-s + (0.911 − 0.911i)7-s + (−0.250 + 0.250i)8-s + (0.253 − 0.967i)9-s + (0.650 − 0.277i)10-s + 0.677i·11-s + (−0.305 − 0.395i)12-s + (0.600 + 0.600i)13-s + 0.911·14-s + (0.272 + 0.962i)15-s − 0.250·16-s + (0.949 + 0.949i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(30\)    =    \(2 \cdot 3 \cdot 5\)
Sign: $0.874 - 0.485i$
Analytic conductor: \(15.4510\)
Root analytic conductor: \(3.93078\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{30} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 30,\ (\ :9/2),\ 0.874 - 0.485i)\)

Particular Values

\(L(5)\) \(\approx\) \(2.16388 + 0.561010i\)
\(L(\frac12)\) \(\approx\) \(2.16388 + 0.561010i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-11.3 - 11.3i)T \)
3 \( 1 + (111. - 85.7i)T \)
5 \( 1 + (-520. + 1.29e3i)T \)
good7 \( 1 + (-5.78e3 + 5.78e3i)T - 4.03e7iT^{2} \)
11 \( 1 - 3.28e4iT - 2.35e9T^{2} \)
13 \( 1 + (-6.18e4 - 6.18e4i)T + 1.06e10iT^{2} \)
17 \( 1 + (-3.26e5 - 3.26e5i)T + 1.18e11iT^{2} \)
19 \( 1 + 6.65e5iT - 3.22e11T^{2} \)
23 \( 1 + (-4.60e5 + 4.60e5i)T - 1.80e12iT^{2} \)
29 \( 1 - 5.52e6T + 1.45e13T^{2} \)
31 \( 1 - 5.23e6T + 2.64e13T^{2} \)
37 \( 1 + (-6.53e6 + 6.53e6i)T - 1.29e14iT^{2} \)
41 \( 1 - 2.26e7iT - 3.27e14T^{2} \)
43 \( 1 + (1.32e7 + 1.32e7i)T + 5.02e14iT^{2} \)
47 \( 1 + (3.43e7 + 3.43e7i)T + 1.11e15iT^{2} \)
53 \( 1 + (-1.24e7 + 1.24e7i)T - 3.29e15iT^{2} \)
59 \( 1 + 8.40e7T + 8.66e15T^{2} \)
61 \( 1 - 1.02e8T + 1.16e16T^{2} \)
67 \( 1 + (-1.62e8 + 1.62e8i)T - 2.72e16iT^{2} \)
71 \( 1 - 2.10e8iT - 4.58e16T^{2} \)
73 \( 1 + (1.03e8 + 1.03e8i)T + 5.88e16iT^{2} \)
79 \( 1 - 6.30e8iT - 1.19e17T^{2} \)
83 \( 1 + (2.94e7 - 2.94e7i)T - 1.86e17iT^{2} \)
89 \( 1 + 2.64e8T + 3.50e17T^{2} \)
97 \( 1 + (4.11e8 - 4.11e8i)T - 7.60e17iT^{2} \)
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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

−15.14441294051606442850042911282, −13.88972430083395565062135408872, −12.60628705992526419523648095947, −11.38113696640108592451571917356, −9.981273994740038596485032711570, −8.367737358917975726618768350766, −6.59491476147983498167477778338, −5.03008858201764521146919875303, −4.23090102402453511094635667239, −1.09402899536860733422847442275, 1.27560062202465805614299549098, 2.85788895689112218435732135330, 5.27604360753761009123664472369, 6.23044485364094402775535306787, 8.026974779959824772623318745543, 10.20429279880852522450415545510, 11.32823830404639602894580093832, 12.09869285327629438549647168946, 13.59809934939701254307017406404, 14.52561013176406996465429746550

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