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Properties

Label 2-30-5.4-c3-0-0
Degree $2$
Conductor $30$
Sign $-0.178 - 0.983i$
Analytic cond. $1.77005$
Root an. cond. $1.33043$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s + 3i·3-s − 4·4-s + (2 + 11i)5-s − 6·6-s − 2i·7-s − 8i·8-s − 9·9-s + (−22 + 4i)10-s + 70·11-s − 12i·12-s − 54i·13-s + 4·14-s + (−33 + 6i)15-s + 16·16-s − 22i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (0.178 + 0.983i)5-s − 0.408·6-s − 0.107i·7-s − 0.353i·8-s − 0.333·9-s + (−0.695 + 0.126i)10-s + 1.91·11-s − 0.288i·12-s − 1.15i·13-s + 0.0763·14-s + (−0.568 + 0.103i)15-s + 0.250·16-s − 0.313i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.178 - 0.983i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.178 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(30\)    =    \(2 \cdot 3 \cdot 5\)
Sign: $-0.178 - 0.983i$
Analytic conductor: \(1.77005\)
Root analytic conductor: \(1.33043\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{30} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 30,\ (\ :3/2),\ -0.178 - 0.983i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.767678 + 0.919842i\)
\(L(\frac12)\) \(\approx\) \(0.767678 + 0.919842i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2iT \)
3 \( 1 - 3iT \)
5 \( 1 + (-2 - 11i)T \)
good7 \( 1 + 2iT - 343T^{2} \)
11 \( 1 - 70T + 1.33e3T^{2} \)
13 \( 1 + 54iT - 2.19e3T^{2} \)
17 \( 1 + 22iT - 4.91e3T^{2} \)
19 \( 1 + 24T + 6.85e3T^{2} \)
23 \( 1 - 100iT - 1.21e4T^{2} \)
29 \( 1 + 216T + 2.43e4T^{2} \)
31 \( 1 - 208T + 2.97e4T^{2} \)
37 \( 1 + 254iT - 5.06e4T^{2} \)
41 \( 1 + 206T + 6.89e4T^{2} \)
43 \( 1 + 292iT - 7.95e4T^{2} \)
47 \( 1 + 320iT - 1.03e5T^{2} \)
53 \( 1 - 402iT - 1.48e5T^{2} \)
59 \( 1 - 370T + 2.05e5T^{2} \)
61 \( 1 + 550T + 2.26e5T^{2} \)
67 \( 1 - 728iT - 3.00e5T^{2} \)
71 \( 1 + 540T + 3.57e5T^{2} \)
73 \( 1 + 604iT - 3.89e5T^{2} \)
79 \( 1 + 792T + 4.93e5T^{2} \)
83 \( 1 + 404iT - 5.71e5T^{2} \)
89 \( 1 - 938T + 7.04e5T^{2} \)
97 \( 1 - 56iT - 9.12e5T^{2} \)
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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

−16.92596340143600097357538620043, −15.42885686336603796306432432538, −14.69583482760204588594300735012, −13.64725299190684495126542792807, −11.72244892902209453657110899494, −10.29804138956363154700175073752, −9.059799341177537628681199805640, −7.25420210453856398975202879064, −5.86132693280654407960530228253, −3.73327048352460643421953506453, 1.51793441717762989104647157434, 4.30556151192794662211149799007, 6.41556334946495839028693815823, 8.563355162756076935360265007437, 9.511412258802902153774109227034, 11.53388052525637704296874415528, 12.28638532742863199342194054136, 13.51502361570863337053377513253, 14.60016700086929928390528059734, 16.66459437335727761570599754612

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