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Properties

Label 2-30-15.8-c9-0-11
Degree $2$
Conductor $30$
Sign $0.701 + 0.712i$
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 + (140. + 0.428i)3-s + 256. i·4-s + (1.19e3 − 728. i)5-s + (−1.58e3 − 1.59e3i)6-s + (454. − 454. i)7-s + (2.89e3 − 2.89e3i)8-s + (1.96e4 + 120. i)9-s + (−2.17e4 − 5.24e3i)10-s + 3.47e4i·11-s + (−109. + 3.59e4i)12-s + (5.18e4 + 5.18e4i)13-s − 1.02e4·14-s + (1.67e5 − 1.01e5i)15-s − 6.55e4·16-s + (1.82e4 + 1.82e4i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.999 + 0.00305i)3-s + 0.500i·4-s + (0.853 − 0.521i)5-s + (−0.498 − 0.501i)6-s + (0.0716 − 0.0716i)7-s + (0.250 − 0.250i)8-s + (0.999 + 0.00610i)9-s + (−0.687 − 0.165i)10-s + 0.714i·11-s + (−0.00152 + 0.499i)12-s + (0.503 + 0.503i)13-s − 0.0716·14-s + (0.854 − 0.518i)15-s − 0.250·16-s + (0.0530 + 0.0530i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(30\)    =    \(2 \cdot 3 \cdot 5\)
Sign: $0.701 + 0.712i$
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: \((2,\ 30,\ (\ :9/2),\ 0.701 + 0.712i)\)

Particular Values

\(L(5)\) \(\approx\) \(2.25222 - 0.943415i\)
\(L(\frac12)\) \(\approx\) \(2.25222 - 0.943415i\)
\(L(\frac{11}{2})\) 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 \( 1 + (-140. - 0.428i)T \)
5 \( 1 + (-1.19e3 + 728. i)T \)
good7 \( 1 + (-454. + 454. i)T - 4.03e7iT^{2} \)
11 \( 1 - 3.47e4iT - 2.35e9T^{2} \)
13 \( 1 + (-5.18e4 - 5.18e4i)T + 1.06e10iT^{2} \)
17 \( 1 + (-1.82e4 - 1.82e4i)T + 1.18e11iT^{2} \)
19 \( 1 + 5.34e5iT - 3.22e11T^{2} \)
23 \( 1 + (-1.78e6 + 1.78e6i)T - 1.80e12iT^{2} \)
29 \( 1 - 2.23e6T + 1.45e13T^{2} \)
31 \( 1 + 5.30e6T + 2.64e13T^{2} \)
37 \( 1 + (6.99e6 - 6.99e6i)T - 1.29e14iT^{2} \)
41 \( 1 + 1.86e7iT - 3.27e14T^{2} \)
43 \( 1 + (-2.52e7 - 2.52e7i)T + 5.02e14iT^{2} \)
47 \( 1 + (2.41e7 + 2.41e7i)T + 1.11e15iT^{2} \)
53 \( 1 + (2.69e7 - 2.69e7i)T - 3.29e15iT^{2} \)
59 \( 1 - 9.15e7T + 8.66e15T^{2} \)
61 \( 1 + 3.23e7T + 1.16e16T^{2} \)
67 \( 1 + (1.74e8 - 1.74e8i)T - 2.72e16iT^{2} \)
71 \( 1 - 3.10e6iT - 4.58e16T^{2} \)
73 \( 1 + (2.47e8 + 2.47e8i)T + 5.88e16iT^{2} \)
79 \( 1 + 1.14e7iT - 1.19e17T^{2} \)
83 \( 1 + (5.23e8 - 5.23e8i)T - 1.86e17iT^{2} \)
89 \( 1 + 2.37e8T + 3.50e17T^{2} \)
97 \( 1 + (8.81e8 - 8.81e8i)T - 7.60e17iT^{2} \)
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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

−14.62589245557328949639319013425, −13.42057489073309940972635884048, −12.52875485807513561241217913114, −10.64893708092340652975774722028, −9.393573501361185203100333382680, −8.630071081071546484087125999552, −6.94983331617244890852570441751, −4.54282781044069671872488969092, −2.61032590847641517086784916531, −1.29290349287178134255046510730, 1.49579283103341073267146043671, 3.22639682955394081277752644267, 5.67697089010037683005937471511, 7.21206317488571519830221733995, 8.569144940701502714129007077969, 9.671493216260335878461266054082, 10.86383278334338520319714817860, 13.10792768721893493690147890126, 14.08400206836169355483942284134, 15.00491454533432919898927464383

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