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Properties

Label 2-375-75.59-c2-0-4
Degree $2$
Conductor $375$
Sign $0.938 - 0.344i$
Analytic cond. $10.2180$
Root an. cond. $3.19656$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.79 − 2.02i)2-s + (1.49 − 2.60i)3-s + (2.44 + 7.52i)4-s + (−9.45 + 4.22i)6-s + 4.79i·7-s + (4.17 − 12.8i)8-s + (−4.52 − 7.78i)9-s + (−3.78 + 5.20i)11-s + (23.2 + 4.90i)12-s + (−10.0 − 13.8i)13-s + (9.72 − 13.3i)14-s + (−12.1 + 8.79i)16-s + (−7.53 + 23.1i)17-s + (−3.15 + 30.9i)18-s + (−2.98 + 9.17i)19-s + ⋯
L(s)  = 1  + (−1.39 − 1.01i)2-s + (0.498 − 0.866i)3-s + (0.611 + 1.88i)4-s + (−1.57 + 0.704i)6-s + 0.684i·7-s + (0.521 − 1.60i)8-s + (−0.502 − 0.864i)9-s + (−0.343 + 0.473i)11-s + (1.93 + 0.408i)12-s + (−0.771 − 1.06i)13-s + (0.694 − 0.956i)14-s + (−0.756 + 0.549i)16-s + (−0.443 + 1.36i)17-s + (−0.175 + 1.71i)18-s + (−0.156 + 0.482i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.344i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.938 - 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $0.938 - 0.344i$
Analytic conductor: \(10.2180\)
Root analytic conductor: \(3.19656\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{375} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 375,\ (\ :1),\ 0.938 - 0.344i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.495380 + 0.0880033i\)
\(L(\frac12)\) \(\approx\) \(0.495380 + 0.0880033i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.49 + 2.60i)T \)
5 \( 1 \)
good2 \( 1 + (2.79 + 2.02i)T + (1.23 + 3.80i)T^{2} \)
7 \( 1 - 4.79iT - 49T^{2} \)
11 \( 1 + (3.78 - 5.20i)T + (-37.3 - 115. i)T^{2} \)
13 \( 1 + (10.0 + 13.8i)T + (-52.2 + 160. i)T^{2} \)
17 \( 1 + (7.53 - 23.1i)T + (-233. - 169. i)T^{2} \)
19 \( 1 + (2.98 - 9.17i)T + (-292. - 212. i)T^{2} \)
23 \( 1 + (-28.8 - 20.9i)T + (163. + 503. i)T^{2} \)
29 \( 1 + (11.9 - 3.89i)T + (680. - 494. i)T^{2} \)
31 \( 1 + (16.0 - 49.3i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (-16.7 - 23.0i)T + (-423. + 1.30e3i)T^{2} \)
41 \( 1 + (9.29 + 12.7i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 + 8.20iT - 1.84e3T^{2} \)
47 \( 1 + (-1.74 - 5.37i)T + (-1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (-0.677 - 2.08i)T + (-2.27e3 + 1.65e3i)T^{2} \)
59 \( 1 + (-46.4 - 63.8i)T + (-1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (13.8 + 10.0i)T + (1.14e3 + 3.53e3i)T^{2} \)
67 \( 1 + (13.6 + 4.43i)T + (3.63e3 + 2.63e3i)T^{2} \)
71 \( 1 + (18.4 - 5.98i)T + (4.07e3 - 2.96e3i)T^{2} \)
73 \( 1 + (-76.5 + 105. i)T + (-1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (5.48 + 16.8i)T + (-5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (10.3 - 31.7i)T + (-5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 + (32.2 - 44.3i)T + (-2.44e3 - 7.53e3i)T^{2} \)
97 \( 1 + (149. - 48.4i)T + (7.61e3 - 5.53e3i)T^{2} \)
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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

−11.03809782021375704936110872401, −10.25853642012987699003216061157, −9.276371855020683663987813415748, −8.565626750989511270403846637615, −7.83316719003680031595318930980, −6.99461232511407924059831678964, −5.50469051115108484863826182688, −3.38703000071678927562225779193, −2.43412913564017031225090977754, −1.39394457153386792489602569275, 0.33983487409906506202640163187, 2.48851093759890294919226191423, 4.30188510111850068660430441377, 5.36910917516214568941133634059, 6.80372054375968176343801482988, 7.44510797582668458416139176378, 8.447689260465620181584744275578, 9.327421057533029555556441118678, 9.688122112328144951136309842546, 10.82593942183625101756959552013

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