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Properties

Label 2-75-15.8-c3-0-12
Degree $2$
Conductor $75$
Sign $0.990 + 0.134i$
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  + (1.39 + 1.39i)2-s + (4.93 − 1.62i)3-s − 4.10i·4-s + (9.15 + 4.61i)6-s + (3.80 − 3.80i)7-s + (16.8 − 16.8i)8-s + (21.7 − 16.0i)9-s + 61.8i·11-s + (−6.67 − 20.2i)12-s + (−48.1 − 48.1i)13-s + 10.6·14-s + 14.3·16-s + (47.5 + 47.5i)17-s + (52.7 + 7.89i)18-s + 93.4i·19-s + ⋯
L(s)  = 1  + (0.493 + 0.493i)2-s + (0.949 − 0.312i)3-s − 0.513i·4-s + (0.623 + 0.314i)6-s + (0.205 − 0.205i)7-s + (0.746 − 0.746i)8-s + (0.804 − 0.594i)9-s + 1.69i·11-s + (−0.160 − 0.487i)12-s + (−1.02 − 1.02i)13-s + 0.202·14-s + 0.223·16-s + (0.679 + 0.679i)17-s + (0.690 + 0.103i)18-s + 1.12i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.134i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.990 + 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.990 + 0.134i$
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: \((2,\ 75,\ (\ :3/2),\ 0.990 + 0.134i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.50148 - 0.168546i\)
\(L(\frac12)\) \(\approx\) \(2.50148 - 0.168546i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.93 + 1.62i)T \)
5 \( 1 \)
good2 \( 1 + (-1.39 - 1.39i)T + 8iT^{2} \)
7 \( 1 + (-3.80 + 3.80i)T - 343iT^{2} \)
11 \( 1 - 61.8iT - 1.33e3T^{2} \)
13 \( 1 + (48.1 + 48.1i)T + 2.19e3iT^{2} \)
17 \( 1 + (-47.5 - 47.5i)T + 4.91e3iT^{2} \)
19 \( 1 - 93.4iT - 6.85e3T^{2} \)
23 \( 1 + (33.7 - 33.7i)T - 1.21e4iT^{2} \)
29 \( 1 + 179.T + 2.43e4T^{2} \)
31 \( 1 + 123.T + 2.97e4T^{2} \)
37 \( 1 + (-10.5 + 10.5i)T - 5.06e4iT^{2} \)
41 \( 1 + 61.8iT - 6.89e4T^{2} \)
43 \( 1 + (133. + 133. i)T + 7.95e4iT^{2} \)
47 \( 1 + (-56.9 - 56.9i)T + 1.03e5iT^{2} \)
53 \( 1 + (-234. + 234. i)T - 1.48e5iT^{2} \)
59 \( 1 - 260.T + 2.05e5T^{2} \)
61 \( 1 - 240.T + 2.26e5T^{2} \)
67 \( 1 + (-320. + 320. i)T - 3.00e5iT^{2} \)
71 \( 1 - 1.08e3iT - 3.57e5T^{2} \)
73 \( 1 + (-208. - 208. i)T + 3.89e5iT^{2} \)
79 \( 1 + 676. iT - 4.93e5T^{2} \)
83 \( 1 + (-71.3 + 71.3i)T - 5.71e5iT^{2} \)
89 \( 1 + 228.T + 7.04e5T^{2} \)
97 \( 1 + (-89.5 + 89.5i)T - 9.12e5iT^{2} \)
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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

−14.44782121257526046310975094229, −13.02635994711118707591629799160, −12.37994780525956590474888673033, −10.25275023887376735800553396483, −9.685025536065567029929818917631, −7.83558036298597397038978447761, −7.12398704374556288382933269071, −5.45296018588763649855991673150, −3.99253064117150469619865735357, −1.81503033441910828879027891400, 2.43421444126157824795225885850, 3.63836623840192012017307902618, 5.03386215011262935759672110049, 7.26571246665362344200299109340, 8.425275168092911859916165837714, 9.400751781164314668032959501411, 10.99207970732846265367017814140, 11.87857834408320279514026628574, 13.20574281388208823857177126533, 13.92858890113835470584097489681

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