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Properties

Label 2-375-25.4-c1-0-13
Degree $2$
Conductor $375$
Sign $0.487 + 0.872i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.487 + 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.487 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $0.487 + 0.872i$
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: \((2,\ 375,\ (\ :1/2),\ 0.487 + 0.872i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65212 - 0.969386i\)
\(L(\frac12)\) \(\approx\) \(1.65212 - 0.969386i\)
\(L(\frac{3}{2})\) 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

−11.23275125055229942909983559744, −10.45287773275949121153395765371, −9.135828091523862132680246960916, −8.528292029100161617216398191225, −7.29344808816614021121523686598, −6.19968848355156299936765400098, −5.55899233099835221684738018014, −4.04228546979646594806571118564, −3.29759583246015784536523362667, −1.12403496442252170139401721677, 2.28632572261205474428495622176, 3.55841470332012526899814740164, 4.45773122365230386577717610740, 5.35841108994159929411591033058, 6.62997408968919675717835676127, 7.948316238110947817115225883261, 9.004835998440758551916604126568, 9.427717284934233803766947179028, 10.87521151259119148725268784650, 11.55949455778242783256330682742

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