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Properties

Label 2-75-15.8-c3-0-14
Degree $2$
Conductor $75$
Sign $-0.931 + 0.364i$
Analytic cond. $4.42514$
Root an. cond. $2.10360$
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  + (−1.18 − 1.18i)2-s + (0.932 − 5.11i)3-s − 5.17i·4-s + (−7.17 + 4.96i)6-s + (13.3 − 13.3i)7-s + (−15.6 + 15.6i)8-s + (−25.2 − 9.53i)9-s + 28.7i·11-s + (−26.4 − 4.83i)12-s + (−14.1 − 14.1i)13-s − 31.7·14-s − 4.25·16-s + (−18.5 − 18.5i)17-s + (18.6 + 41.3i)18-s − 49.0i·19-s + ⋯
L(s)  = 1  + (−0.419 − 0.419i)2-s + (0.179 − 0.983i)3-s − 0.647i·4-s + (−0.488 + 0.337i)6-s + (0.721 − 0.721i)7-s + (−0.691 + 0.691i)8-s + (−0.935 − 0.353i)9-s + 0.787i·11-s + (−0.636 − 0.116i)12-s + (−0.302 − 0.302i)13-s − 0.605·14-s − 0.0664·16-s + (−0.264 − 0.264i)17-s + (0.244 + 0.541i)18-s − 0.592i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.931 + 0.364i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.931 + 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.931 + 0.364i$
Analytic conductor: \(4.42514\)
Root analytic conductor: \(2.10360\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :3/2),\ -0.931 + 0.364i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.208197 - 1.10314i\)
\(L(\frac12)\) \(\approx\) \(0.208197 - 1.10314i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.932 + 5.11i)T \)
5 \( 1 \)
good2 \( 1 + (1.18 + 1.18i)T + 8iT^{2} \)
7 \( 1 + (-13.3 + 13.3i)T - 343iT^{2} \)
11 \( 1 - 28.7iT - 1.33e3T^{2} \)
13 \( 1 + (14.1 + 14.1i)T + 2.19e3iT^{2} \)
17 \( 1 + (18.5 + 18.5i)T + 4.91e3iT^{2} \)
19 \( 1 + 49.0iT - 6.85e3T^{2} \)
23 \( 1 + (-37.7 + 37.7i)T - 1.21e4iT^{2} \)
29 \( 1 - 125.T + 2.43e4T^{2} \)
31 \( 1 - 247.T + 2.97e4T^{2} \)
37 \( 1 + (-127. + 127. i)T - 5.06e4iT^{2} \)
41 \( 1 + 390. iT - 6.89e4T^{2} \)
43 \( 1 + (-39.3 - 39.3i)T + 7.95e4iT^{2} \)
47 \( 1 + (124. + 124. i)T + 1.03e5iT^{2} \)
53 \( 1 + (-160. + 160. i)T - 1.48e5iT^{2} \)
59 \( 1 + 729.T + 2.05e5T^{2} \)
61 \( 1 - 2T + 2.26e5T^{2} \)
67 \( 1 + (-329. + 329. i)T - 3.00e5iT^{2} \)
71 \( 1 - 171. iT - 3.57e5T^{2} \)
73 \( 1 + (-279. - 279. i)T + 3.89e5iT^{2} \)
79 \( 1 + 48.0iT - 4.93e5T^{2} \)
83 \( 1 + (144. - 144. i)T - 5.71e5iT^{2} \)
89 \( 1 - 1.41e3T + 7.04e5T^{2} \)
97 \( 1 + (908. - 908. i)T - 9.12e5iT^{2} \)
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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

−13.66162208762178973938185557699, −12.34242169013754562911415349532, −11.31349348280292438982109773423, −10.29751043100024111105049185130, −8.990449943206975323959456608043, −7.74406648019223650536074587681, −6.56613746938243289592000115940, −4.91288570277976723812451043895, −2.35456002458883041322480859123, −0.835816795275370201792001187246, 2.98666347436891069574896718930, 4.58537515729170967344608694103, 6.15650483179051940131039844169, 8.025231426049988264520307317857, 8.667016772988260509748904231887, 9.776400365778931262893402581715, 11.23040481911644280142956287305, 12.09890181322226665237453757289, 13.65140290484601497752852215921, 14.83308077814925462065484306755

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