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Properties

Label 2-375-25.19-c1-0-5
Degree $2$
Conductor $375$
Sign $0.957 - 0.288i$
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  + (−1.46 + 0.474i)2-s + (−0.587 − 0.809i)3-s + (0.292 − 0.212i)4-s + (1.24 + 0.903i)6-s + 1.49i·7-s + (1.48 − 2.03i)8-s + (−0.309 + 0.951i)9-s + (−0.728 − 2.24i)11-s + (−0.343 − 0.111i)12-s + (−1.28 − 0.417i)13-s + (−0.710 − 2.18i)14-s + (−1.41 + 4.36i)16-s + (−1.28 + 1.77i)17-s − 1.53i·18-s + (4.62 + 3.35i)19-s + ⋯
L(s)  = 1  + (−1.03 + 0.335i)2-s + (−0.339 − 0.467i)3-s + (0.146 − 0.106i)4-s + (0.507 + 0.368i)6-s + 0.565i·7-s + (0.523 − 0.720i)8-s + (−0.103 + 0.317i)9-s + (−0.219 − 0.675i)11-s + (−0.0991 − 0.0322i)12-s + (−0.355 − 0.115i)13-s + (−0.189 − 0.584i)14-s + (−0.354 + 1.09i)16-s + (−0.312 + 0.430i)17-s − 0.362i·18-s + (1.05 + 0.770i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $0.957 - 0.288i$
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: \((2,\ 375,\ (\ :1/2),\ 0.957 - 0.288i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.643647 + 0.0949970i\)
\(L(\frac12)\) \(\approx\) \(0.643647 + 0.0949970i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.46 - 0.474i)T + (1.61 - 1.17i)T^{2} \)
7 \( 1 - 1.49iT - 7T^{2} \)
11 \( 1 + (0.728 + 2.24i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (1.28 + 0.417i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.28 - 1.77i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-4.62 - 3.35i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-8.36 + 2.71i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (-6.39 + 4.64i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.99 - 2.17i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-9.27 - 3.01i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.573 + 1.76i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 8.01iT - 43T^{2} \)
47 \( 1 + (3.91 + 5.39i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.45 - 3.37i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (3.41 - 10.5i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (3.78 + 11.6i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-2.53 + 3.48i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (4.67 - 3.39i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-6.58 + 2.14i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (8.63 - 6.27i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (0.131 - 0.181i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + (0.132 + 0.408i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (7.34 + 10.1i)T + (-29.9 + 92.2i)T^{2} \)
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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

−11.29366925698587476322491879808, −10.35016297614042765792848787207, −9.456816339531232762566652448076, −8.500942413118158101581457347908, −7.891376976694241442772783785613, −6.84743683984871070315716771627, −5.91086761409996578805084579295, −4.64100023416732267791404719457, −2.90091537740404515727710755309, −1.00195974655183484946386372742, 0.936611086000415363965316519826, 2.78194516388211435344380493303, 4.54112448714314311549427912446, 5.22740434603380531327373114931, 6.92154411659456453220403734264, 7.64781955065891836195423743076, 8.952454789188705369757988455011, 9.535395921475948891798362634653, 10.31826691356941393791394528709, 11.08797335418661953697837893060

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