Properties

Label 2-75-5.3-c4-0-11
Degree $2$
Conductor $75$
Sign $-0.991 + 0.130i$
Analytic cond. $7.75274$
Root an. cond. $2.78437$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.44 − 5.44i)2-s + (−3.67 − 3.67i)3-s − 43.3i·4-s − 40.0·6-s + (−19.2 + 19.2i)7-s + (−149. − 149. i)8-s + 27i·9-s + 185.·11-s + (−159. + 159. i)12-s + (−48.5 − 48.5i)13-s + 209. i·14-s − 932.·16-s + (147. − 147. i)17-s + (147. + 147. i)18-s − 140. i·19-s + ⋯
L(s)  = 1  + (1.36 − 1.36i)2-s + (−0.408 − 0.408i)3-s − 2.71i·4-s − 1.11·6-s + (−0.392 + 0.392i)7-s + (−2.33 − 2.33i)8-s + 0.333i·9-s + 1.52·11-s + (−1.10 + 1.10i)12-s + (−0.287 − 0.287i)13-s + 1.06i·14-s − 3.64·16-s + (0.510 − 0.510i)17-s + (0.454 + 0.454i)18-s − 0.388i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.130i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-0.991 + 0.130i$
Analytic conductor: \(7.75274\)
Root analytic conductor: \(2.78437\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :2),\ -0.991 + 0.130i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.177699 - 2.70608i\)
\(L(\frac12)\) \(\approx\) \(0.177699 - 2.70608i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.67 + 3.67i)T \)
5 \( 1 \)
good2 \( 1 + (-5.44 + 5.44i)T - 16iT^{2} \)
7 \( 1 + (19.2 - 19.2i)T - 2.40e3iT^{2} \)
11 \( 1 - 185.T + 1.46e4T^{2} \)
13 \( 1 + (48.5 + 48.5i)T + 2.85e4iT^{2} \)
17 \( 1 + (-147. + 147. i)T - 8.35e4iT^{2} \)
19 \( 1 + 140. iT - 1.30e5T^{2} \)
23 \( 1 + (177. + 177. i)T + 2.79e5iT^{2} \)
29 \( 1 - 588. iT - 7.07e5T^{2} \)
31 \( 1 - 1.41e3T + 9.23e5T^{2} \)
37 \( 1 + (-763. + 763. i)T - 1.87e6iT^{2} \)
41 \( 1 - 995.T + 2.82e6T^{2} \)
43 \( 1 + (657. + 657. i)T + 3.41e6iT^{2} \)
47 \( 1 + (102. - 102. i)T - 4.87e6iT^{2} \)
53 \( 1 + (365. + 365. i)T + 7.89e6iT^{2} \)
59 \( 1 - 5.80e3iT - 1.21e7T^{2} \)
61 \( 1 - 6.28e3T + 1.38e7T^{2} \)
67 \( 1 + (5.49e3 - 5.49e3i)T - 2.01e7iT^{2} \)
71 \( 1 + 7.66e3T + 2.54e7T^{2} \)
73 \( 1 + (-6.40e3 - 6.40e3i)T + 2.83e7iT^{2} \)
79 \( 1 + 5.69e3iT - 3.89e7T^{2} \)
83 \( 1 + (-999. - 999. i)T + 4.74e7iT^{2} \)
89 \( 1 + 3.09e3iT - 6.27e7T^{2} \)
97 \( 1 + (5.03e3 - 5.03e3i)T - 8.85e7iT^{2} \)
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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

−13.02698841033854282035988967889, −12.08987079313494762770058899461, −11.60504159007760871510264953290, −10.31793173930011586830145308359, −9.225445194646510880649149875637, −6.69487255781969032929915609585, −5.64029571475224168860075105842, −4.25636573338282233567585744074, −2.74264031121707381870176392593, −1.06088923561379730545613837749, 3.57961013172137567669517708411, 4.52414086184048116768220217979, 6.01421513174314652331815677360, 6.75422491142349888577509992525, 8.138232126309990311147531922357, 9.644879069984226023507065769477, 11.59717860993632456267813164372, 12.33378378899175960490479015243, 13.56742302019241157407186544565, 14.42932031762869175583211950557

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