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Properties

Label 2-375-25.9-c1-0-6
Degree $2$
Conductor $375$
Sign $0.617 - 0.786i$
Analytic cond. $2.99439$
Root an. cond. $1.73043$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.00 + 1.38i)2-s + (0.951 − 0.309i)3-s + (−0.281 − 0.867i)4-s + (−0.527 + 1.62i)6-s − 3.94i·7-s + (−1.76 − 0.573i)8-s + (0.809 − 0.587i)9-s + (4.78 + 3.47i)11-s + (−0.535 − 0.737i)12-s + (1.93 + 2.66i)13-s + (5.44 + 3.95i)14-s + (4.03 − 2.93i)16-s + (2.57 + 0.836i)17-s + 1.70i·18-s + (0.728 − 2.24i)19-s + ⋯
L(s)  = 1  + (−0.709 + 0.976i)2-s + (0.549 − 0.178i)3-s + (−0.140 − 0.433i)4-s + (−0.215 + 0.662i)6-s − 1.49i·7-s + (−0.624 − 0.202i)8-s + (0.269 − 0.195i)9-s + (1.44 + 1.04i)11-s + (−0.154 − 0.212i)12-s + (0.536 + 0.738i)13-s + (1.45 + 1.05i)14-s + (1.00 − 0.733i)16-s + (0.624 + 0.202i)17-s + 0.402i·18-s + (0.167 − 0.514i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $0.617 - 0.786i$
Analytic conductor: \(2.99439\)
Root analytic conductor: \(1.73043\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{375} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 375,\ (\ :1/2),\ 0.617 - 0.786i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09440 + 0.532206i\)
\(L(\frac12)\) \(\approx\) \(1.09440 + 0.532206i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.951 + 0.309i)T \)
5 \( 1 \)
good2 \( 1 + (1.00 - 1.38i)T + (-0.618 - 1.90i)T^{2} \)
7 \( 1 + 3.94iT - 7T^{2} \)
11 \( 1 + (-4.78 - 3.47i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-1.93 - 2.66i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.57 - 0.836i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-0.728 + 2.24i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-0.343 + 0.472i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.20 - 3.72i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.837 + 2.57i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-0.0122 - 0.0168i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (1.19 - 0.865i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 1.27iT - 43T^{2} \)
47 \( 1 + (5.16 - 1.67i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-2.67 + 0.870i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (3.79 - 2.75i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (4.51 + 3.28i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-5.73 - 1.86i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (2.50 + 7.70i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (7.82 - 10.7i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (5.14 + 15.8i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.743 + 0.241i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (2.80 + 2.04i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-2.33 + 0.758i)T + (78.4 - 57.0i)T^{2} \)
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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

−11.48878507141038753626290847452, −10.18753785326810744753991202810, −9.434956525363061636980310259755, −8.675014470256005140585570586522, −7.57149828189901773049760882188, −7.01674299997089866675287631043, −6.34527397492249428071566714754, −4.44943534859283540928302789469, −3.50136912094147932017396322805, −1.32282422521190531723392171951, 1.36175513843460135232728773272, 2.77258990476836687068383310607, 3.59979774303519029171141236830, 5.52907267379517359862903385098, 6.28303241855506342064366830215, 8.132431307587918844648229764939, 8.736896443833649694269276596264, 9.366092648170401779882478861444, 10.20985523334373852104181449678, 11.31242286922705420139653621591

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