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Properties

Label 2-375-25.16-c1-0-10
Degree $2$
Conductor $375$
Sign $0.988 + 0.152i$
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  + (1.24 + 0.903i)2-s + (−0.309 − 0.951i)3-s + (0.112 + 0.346i)4-s + (0.475 − 1.46i)6-s + 1.68·7-s + (0.777 − 2.39i)8-s + (−0.809 + 0.587i)9-s + (2.40 + 1.75i)11-s + (0.294 − 0.214i)12-s + (−0.188 + 0.136i)13-s + (2.09 + 1.52i)14-s + (3.71 − 2.70i)16-s + (2.30 − 7.09i)17-s − 1.53·18-s + (0.232 − 0.716i)19-s + ⋯
L(s)  = 1  + (0.879 + 0.639i)2-s + (−0.178 − 0.549i)3-s + (0.0563 + 0.173i)4-s + (0.193 − 0.597i)6-s + 0.637·7-s + (0.274 − 0.845i)8-s + (−0.269 + 0.195i)9-s + (0.726 + 0.527i)11-s + (0.0851 − 0.0618i)12-s + (−0.0522 + 0.0379i)13-s + (0.560 + 0.407i)14-s + (0.929 − 0.675i)16-s + (0.559 − 1.72i)17-s − 0.362·18-s + (0.0534 − 0.164i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.152i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $0.988 + 0.152i$
Analytic conductor: \(2.99439\)
Root analytic conductor: \(1.73043\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{375} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 375,\ (\ :1/2),\ 0.988 + 0.152i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.10387 - 0.161119i\)
\(L(\frac12)\) \(\approx\) \(2.10387 - 0.161119i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 \)
good2 \( 1 + (-1.24 - 0.903i)T + (0.618 + 1.90i)T^{2} \)
7 \( 1 - 1.68T + 7T^{2} \)
11 \( 1 + (-2.40 - 1.75i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.188 - 0.136i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.30 + 7.09i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.232 + 0.716i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (0.706 + 0.512i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-2.12 - 6.53i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (3.03 - 9.33i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (8.19 - 5.95i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (3.07 - 2.23i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 5.27T + 43T^{2} \)
47 \( 1 + (2.64 + 8.14i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-1.84 - 5.68i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-3.11 + 2.26i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-3.55 - 2.58i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-0.554 + 1.70i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (1.35 + 4.16i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-12.1 - 8.84i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-2.27 - 7.01i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-1.34 + 4.13i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (9.79 + 7.11i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-2.92 - 9.01i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.75963225442794451375967498508, −10.50827915850333789043011454352, −9.501270395069631580918604648226, −8.357023243406811832633366118423, −6.98707961843035178697755038785, −6.85352192461675030086834094202, −5.26908028639268378989835584832, −4.87278680594464180879815772703, −3.34172124943564404375541833756, −1.40529539634114284138624195837, 1.90894565323441361854961969142, 3.55154763885154960831362310818, 4.14910102786362086955630330015, 5.33134769559719159189274440774, 6.18299584390689825357742808203, 7.888949555092600739987132133760, 8.563788941777051966076679664196, 9.833345451467995198693907913709, 10.81752986368680918810723409147, 11.46002683912293720303889412359

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