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Properties

Label 2-375-25.19-c1-0-13
Degree $2$
Conductor $375$
Sign $0.852 + 0.522i$
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  + (2.23 − 0.726i)2-s + (0.587 + 0.809i)3-s + (2.85 − 2.07i)4-s + (1.90 + 1.38i)6-s − 3.48i·7-s + (2.10 − 2.90i)8-s + (−0.309 + 0.951i)9-s + (0.905 + 2.78i)11-s + (3.35 + 1.08i)12-s + (−1.78 − 0.579i)13-s + (−2.52 − 7.78i)14-s + (0.427 − 1.31i)16-s + (−3.98 + 5.48i)17-s + 2.35i·18-s + (−2.38 − 1.73i)19-s + ⋯
L(s)  = 1  + (1.58 − 0.513i)2-s + (0.339 + 0.467i)3-s + (1.42 − 1.03i)4-s + (0.776 + 0.564i)6-s − 1.31i·7-s + (0.745 − 1.02i)8-s + (−0.103 + 0.317i)9-s + (0.273 + 0.840i)11-s + (0.968 + 0.314i)12-s + (−0.494 − 0.160i)13-s + (−0.676 − 2.08i)14-s + (0.106 − 0.328i)16-s + (−0.967 + 1.33i)17-s + 0.554i·18-s + (−0.547 − 0.397i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $0.852 + 0.522i$
Analytic conductor: \(2.99439\)
Root analytic conductor: \(1.73043\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{375} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 375,\ (\ :1/2),\ 0.852 + 0.522i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.18208 - 0.896470i\)
\(L(\frac12)\) \(\approx\) \(3.18208 - 0.896470i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.587 - 0.809i)T \)
5 \( 1 \)
good2 \( 1 + (-2.23 + 0.726i)T + (1.61 - 1.17i)T^{2} \)
7 \( 1 + 3.48iT - 7T^{2} \)
11 \( 1 + (-0.905 - 2.78i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (1.78 + 0.579i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (3.98 - 5.48i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (2.38 + 1.73i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-5.22 + 1.69i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (-2.06 + 1.50i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.338 + 0.245i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (4.98 + 1.61i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.518 + 1.59i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 10.9iT - 43T^{2} \)
47 \( 1 + (4.40 + 6.06i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.18 + 3.00i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.19 + 6.76i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.98 + 6.12i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (5.90 - 8.12i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (0.589 - 0.428i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.41 + 1.11i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-2.48 + 1.80i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-5.94 + 8.18i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + (0.0888 + 0.273i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (6.11 + 8.42i)T + (-29.9 + 92.2i)T^{2} \)
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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

−11.25052610832888998858477352063, −10.67531121422962050353218923294, −9.892069360291630528571826703803, −8.543560750176325763922601755990, −7.16602413385287382958625608275, −6.36079968084353708428776214590, −4.81924865195961396698537460784, −4.34514232167690404372519353436, −3.37883161757150170857997870823, −1.99345730680267250533415107793, 2.39837785864715169791562485233, 3.26794772081000876668988339546, 4.72247939693117477747343053731, 5.58518413507133357769256355563, 6.49717813135420127906200440114, 7.28671509640646502662628091412, 8.601920534119109905641443221182, 9.290866916266676490476613239753, 11.06923721895152565371549659677, 11.93701914751816466747148048346

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