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Properties

Label 2-30-15.8-c3-0-4
Degree $2$
Conductor $30$
Sign $0.996 - 0.0795i$
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  + (1.41 + 1.41i)2-s + (2.01 − 4.79i)3-s + 4.00i·4-s + (10.8 − 2.71i)5-s + (9.62 − 3.93i)6-s + (−16.1 + 16.1i)7-s + (−5.65 + 5.65i)8-s + (−18.9 − 19.2i)9-s + (19.1 + 11.4i)10-s + 12.7i·11-s + (19.1 + 8.04i)12-s + (−34.2 − 34.2i)13-s − 45.7·14-s + (8.80 − 57.4i)15-s − 16.0·16-s + (−26.9 − 26.9i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.387 − 0.922i)3-s + 0.500i·4-s + (0.970 − 0.242i)5-s + (0.654 − 0.267i)6-s + (−0.873 + 0.873i)7-s + (−0.250 + 0.250i)8-s + (−0.700 − 0.713i)9-s + (0.606 + 0.363i)10-s + 0.349i·11-s + (0.461 + 0.193i)12-s + (−0.729 − 0.729i)13-s − 0.873·14-s + (0.151 − 0.988i)15-s − 0.250·16-s + (−0.385 − 0.385i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(30\)    =    \(2 \cdot 3 \cdot 5\)
Sign: $0.996 - 0.0795i$
Analytic conductor: \(1.77005\)
Root analytic conductor: \(1.33043\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{30} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 30,\ (\ :3/2),\ 0.996 - 0.0795i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.68461 + 0.0671454i\)
\(L(\frac12)\) \(\approx\) \(1.68461 + 0.0671454i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 - 1.41i)T \)
3 \( 1 + (-2.01 + 4.79i)T \)
5 \( 1 + (-10.8 + 2.71i)T \)
good7 \( 1 + (16.1 - 16.1i)T - 343iT^{2} \)
11 \( 1 - 12.7iT - 1.33e3T^{2} \)
13 \( 1 + (34.2 + 34.2i)T + 2.19e3iT^{2} \)
17 \( 1 + (26.9 + 26.9i)T + 4.91e3iT^{2} \)
19 \( 1 - 136. iT - 6.85e3T^{2} \)
23 \( 1 + (-72.2 + 72.2i)T - 1.21e4iT^{2} \)
29 \( 1 - 206.T + 2.43e4T^{2} \)
31 \( 1 - 6.41T + 2.97e4T^{2} \)
37 \( 1 + (-148. + 148. i)T - 5.06e4iT^{2} \)
41 \( 1 + 20.5iT - 6.89e4T^{2} \)
43 \( 1 + (162. + 162. i)T + 7.95e4iT^{2} \)
47 \( 1 + (191. + 191. i)T + 1.03e5iT^{2} \)
53 \( 1 + (12.0 - 12.0i)T - 1.48e5iT^{2} \)
59 \( 1 - 627.T + 2.05e5T^{2} \)
61 \( 1 + 543.T + 2.26e5T^{2} \)
67 \( 1 + (120. - 120. i)T - 3.00e5iT^{2} \)
71 \( 1 - 149. iT - 3.57e5T^{2} \)
73 \( 1 + (-771. - 771. i)T + 3.89e5iT^{2} \)
79 \( 1 - 97.9iT - 4.93e5T^{2} \)
83 \( 1 + (735. - 735. i)T - 5.71e5iT^{2} \)
89 \( 1 + 492.T + 7.04e5T^{2} \)
97 \( 1 + (811. - 811. i)T - 9.12e5iT^{2} \)
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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

−16.55601099607023462419936347826, −15.07805126710960768637971805364, −14.04074142665503745325473995516, −12.81480314299627738915815237343, −12.30382254454163735358445872505, −9.787711516272582838873154431809, −8.439461367194242531913784905324, −6.74095255053762457048949478101, −5.59079399578043611699415218112, −2.63684247479145437189268047591, 2.93278997155875450581078334757, 4.73115582973570092952389375873, 6.62191064057652483505732774680, 9.218593103071907653509060027577, 10.08775028545774684948547865345, 11.18720897991430593441164899229, 13.22047271361440161110556311863, 13.88170840845292537187560072440, 15.09075183732822740485958123202, 16.43478205012298230909705287400

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