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Properties

Label 2-375-25.6-c1-0-9
Degree $2$
Conductor $375$
Sign $0.0627 + 0.998i$
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  + (−0.309 − 0.951i)2-s + (0.809 − 0.587i)3-s + (0.809 − 0.587i)4-s + (−0.809 − 0.587i)6-s + 4.47·7-s + (−2.42 − 1.76i)8-s + (0.309 − 0.951i)9-s + (−0.381 − 1.17i)11-s + (0.309 − 0.951i)12-s + (−1.73 + 5.34i)13-s + (−1.38 − 4.25i)14-s + (−0.309 + 0.951i)16-s + (3.11 + 2.26i)17-s − 18-s + (−1 − 0.726i)19-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (0.467 − 0.339i)3-s + (0.404 − 0.293i)4-s + (−0.330 − 0.239i)6-s + 1.69·7-s + (−0.858 − 0.623i)8-s + (0.103 − 0.317i)9-s + (−0.115 − 0.354i)11-s + (0.0892 − 0.274i)12-s + (−0.481 + 1.48i)13-s + (−0.369 − 1.13i)14-s + (−0.0772 + 0.237i)16-s + (0.756 + 0.549i)17-s − 0.235·18-s + (−0.229 − 0.166i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0627 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0627 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $0.0627 + 0.998i$
Analytic conductor: \(2.99439\)
Root analytic conductor: \(1.73043\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{375} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 375,\ (\ :1/2),\ 0.0627 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27305 - 1.19547i\)
\(L(\frac12)\) \(\approx\) \(1.27305 - 1.19547i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 \)
good2 \( 1 + (0.309 + 0.951i)T + (-1.61 + 1.17i)T^{2} \)
7 \( 1 - 4.47T + 7T^{2} \)
11 \( 1 + (0.381 + 1.17i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (1.73 - 5.34i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-3.11 - 2.26i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1 + 0.726i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (1.38 + 4.25i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (5.35 - 3.88i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.23 + 1.62i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.954 + 2.93i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.11 - 3.44i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 7.70T + 43T^{2} \)
47 \( 1 + (0.618 - 0.449i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-2.92 + 2.12i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.23 - 3.80i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-0.5 - 1.53i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-0.618 - 0.449i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-4.23 + 3.07i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-2.5 - 7.69i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-10.0 - 7.33i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-1.66 - 5.11i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-1.73 + 1.26i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.25552003345125837238977687319, −10.42425403251349202903988253782, −9.326460929635996813690675063161, −8.474645485912936486963333364177, −7.51827264882841634744818982390, −6.49701876756438252216868774570, −5.20159845020355634082914399363, −3.90058934184434189992978776609, −2.29712323887174060236925048768, −1.49452257119849312698406121408, 2.03586946552576355696685847344, 3.38486713852217235069959721210, 4.95663508661845574926060382666, 5.66740976391047744465236655478, 7.34565977069174786116046211255, 7.83224391203375347823281339148, 8.437302965450371133319213844978, 9.659668154788361565227618591603, 10.68469364952262928512394780458, 11.55914444959464475932967016185

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