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Properties

Label 2-375-25.14-c1-0-5
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: $\chi_{375} (199, \cdot )$
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

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

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