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Properties

Label 2-375-25.4-c1-0-9
Degree $2$
Conductor $375$
Sign $-0.998 + 0.0458i$
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  + (−1.06 − 0.346i)2-s + (−0.587 + 0.809i)3-s + (−0.598 − 0.435i)4-s + (0.907 − 0.659i)6-s + 1.11i·7-s + (1.80 + 2.48i)8-s + (−0.309 − 0.951i)9-s + (1.13 − 3.49i)11-s + (0.704 − 0.228i)12-s + (−3.85 + 1.25i)13-s + (0.386 − 1.18i)14-s + (−0.609 − 1.87i)16-s + (−1.24 − 1.71i)17-s + 1.12i·18-s + (−3.28 + 2.38i)19-s + ⋯
L(s)  = 1  + (−0.754 − 0.245i)2-s + (−0.339 + 0.467i)3-s + (−0.299 − 0.217i)4-s + (0.370 − 0.269i)6-s + 0.420i·7-s + (0.639 + 0.879i)8-s + (−0.103 − 0.317i)9-s + (0.341 − 1.05i)11-s + (0.203 − 0.0660i)12-s + (−1.06 + 0.347i)13-s + (0.103 − 0.317i)14-s + (−0.152 − 0.468i)16-s + (−0.302 − 0.416i)17-s + 0.264i·18-s + (−0.753 + 0.547i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $-0.998 + 0.0458i$
Analytic conductor: \(2.99439\)
Root analytic conductor: \(1.73043\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{375} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 375,\ (\ :1/2),\ -0.998 + 0.0458i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.000825525 - 0.0359571i\)
\(L(\frac12)\) \(\approx\) \(0.000825525 - 0.0359571i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.06 + 0.346i)T + (1.61 + 1.17i)T^{2} \)
7 \( 1 - 1.11iT - 7T^{2} \)
11 \( 1 + (-1.13 + 3.49i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (3.85 - 1.25i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.24 + 1.71i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (3.28 - 2.38i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (5.87 + 1.90i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (1.82 + 1.32i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (8.13 - 5.90i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (7.01 - 2.27i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.30 + 7.10i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 9.24iT - 43T^{2} \)
47 \( 1 + (1.83 - 2.53i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-2.06 + 2.83i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.03 - 6.27i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.81 - 8.65i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (1.54 + 2.12i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (0.534 + 0.388i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (7.10 + 2.31i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-6.90 - 5.01i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-7.20 - 9.92i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (-4.72 + 14.5i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-7.93 + 10.9i)T + (-29.9 - 92.2i)T^{2} \)
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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

−10.63828667877869440596309720820, −10.12004975190401476548724532483, −8.981070504538310623031579598233, −8.647581630422406875521370709011, −7.27870139870328992221146500860, −5.92316679098689606046278366947, −5.07856908800585186900243770016, −3.85422271269620218188624283918, −2.06877065029130923020766829491, −0.03126363320247873490857588680, 1.92249003982293829736214705525, 3.92734317461072049581738868484, 4.94031374988798838969257472673, 6.45006079710147426944071283331, 7.36416000994335514270017700703, 7.901226060291785107350692609726, 9.119764376773092347016460269569, 9.874181150897113709983841350565, 10.69519375151904980257000905303, 11.89293561251921482278957956786

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