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Properties

Label 2-30-15.8-c1-0-1
Degree $2$
Conductor $30$
Sign $0.662 + 0.749i$
Analytic cond. $0.239551$
Root an. cond. $0.489439$
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  + (−0.707 − 0.707i)2-s + (−0.292 − 1.70i)3-s + 1.00i·4-s + (0.707 + 2.12i)5-s + (−0.999 + 1.41i)6-s + (−1 + i)7-s + (0.707 − 0.707i)8-s + (−2.82 + i)9-s + (0.999 − 2i)10-s − 1.41i·11-s + (1.70 − 0.292i)12-s + 1.41·14-s + (3.41 − 1.82i)15-s − 1.00·16-s + (−1.41 − 1.41i)17-s + (2.70 + 1.29i)18-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.169 − 0.985i)3-s + 0.500i·4-s + (0.316 + 0.948i)5-s + (−0.408 + 0.577i)6-s + (−0.377 + 0.377i)7-s + (0.250 − 0.250i)8-s + (−0.942 + 0.333i)9-s + (0.316 − 0.632i)10-s − 0.426i·11-s + (0.492 − 0.0845i)12-s + 0.377·14-s + (0.881 − 0.472i)15-s − 0.250·16-s + (−0.342 − 0.342i)17-s + (0.638 + 0.304i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(30\)    =    \(2 \cdot 3 \cdot 5\)
Sign: $0.662 + 0.749i$
Analytic conductor: \(0.239551\)
Root analytic conductor: \(0.489439\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{30} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 30,\ (\ :1/2),\ 0.662 + 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.500941 - 0.225904i\)
\(L(\frac12)\) \(\approx\) \(0.500941 - 0.225904i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 + (0.292 + 1.70i)T \)
5 \( 1 + (-0.707 - 2.12i)T \)
good7 \( 1 + (1 - i)T - 7iT^{2} \)
11 \( 1 + 1.41iT - 11T^{2} \)
13 \( 1 + 13iT^{2} \)
17 \( 1 + (1.41 + 1.41i)T + 17iT^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + (2.82 - 2.82i)T - 23iT^{2} \)
29 \( 1 - 7.07T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + (-6 + 6i)T - 37iT^{2} \)
41 \( 1 + 5.65iT - 41T^{2} \)
43 \( 1 + (-6 - 6i)T + 43iT^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 + (-2.82 + 2.82i)T - 53iT^{2} \)
59 \( 1 + 9.89T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + (4 - 4i)T - 67iT^{2} \)
71 \( 1 - 14.1iT - 71T^{2} \)
73 \( 1 + (5 + 5i)T + 73iT^{2} \)
79 \( 1 + 6iT - 79T^{2} \)
83 \( 1 + (-8.48 + 8.48i)T - 83iT^{2} \)
89 \( 1 + 2.82T + 89T^{2} \)
97 \( 1 + (-3 + 3i)T - 97iT^{2} \)
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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

−17.48307595123005864603652837595, −15.89890463846466991816436824987, −14.20126631705845841631271124206, −13.14067675765468224873364350630, −11.80062970170223287271475144603, −10.76674121723936565168200091958, −9.142985922861602836403785620908, −7.49688463317062462469160487759, −6.16709329434069627483400906865, −2.68850278398652390264720053394, 4.50368004225802633236355559239, 6.10595906160915434301636909297, 8.279743490344514498393476746399, 9.526070664807416336095550719717, 10.45471285693397489650266968309, 12.23668530395967147757095350018, 13.85289691093411225209009374022, 15.19197698537520675261550340977, 16.32141304116390009919213512709, 16.90375285714291204555643085105

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