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Properties

Label 2-375-25.19-c1-0-12
Degree $2$
Conductor $375$
Sign $-0.714 + 0.699i$
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  + (0.510 − 0.165i)2-s + (−0.587 − 0.809i)3-s + (−1.38 + 1.00i)4-s + (−0.434 − 0.315i)6-s − 2.57i·7-s + (−1.17 + 1.61i)8-s + (−0.309 + 0.951i)9-s + (−1.58 − 4.89i)11-s + (1.62 + 0.529i)12-s + (−1.40 − 0.455i)13-s + (−0.426 − 1.31i)14-s + (0.727 − 2.24i)16-s + (−0.404 + 0.556i)17-s + 0.536i·18-s + (−6.54 − 4.75i)19-s + ⋯
L(s)  = 1  + (0.360 − 0.117i)2-s + (−0.339 − 0.467i)3-s + (−0.692 + 0.503i)4-s + (−0.177 − 0.128i)6-s − 0.972i·7-s + (−0.413 + 0.569i)8-s + (−0.103 + 0.317i)9-s + (−0.479 − 1.47i)11-s + (0.470 + 0.152i)12-s + (−0.389 − 0.126i)13-s + (−0.114 − 0.350i)14-s + (0.181 − 0.560i)16-s + (−0.0980 + 0.134i)17-s + 0.126i·18-s + (−1.50 − 1.09i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.714 + 0.699i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.714 + 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $-0.714 + 0.699i$
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: \((2,\ 375,\ (\ :1/2),\ -0.714 + 0.699i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.269471 - 0.660179i\)
\(L(\frac12)\) \(\approx\) \(0.269471 - 0.660179i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.510 + 0.165i)T + (1.61 - 1.17i)T^{2} \)
7 \( 1 + 2.57iT - 7T^{2} \)
11 \( 1 + (1.58 + 4.89i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (1.40 + 0.455i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.404 - 0.556i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (6.54 + 4.75i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (0.354 - 0.115i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (-0.0288 + 0.0209i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-3.63 - 2.63i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.81 + 0.590i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.59 - 4.91i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 11.4iT - 43T^{2} \)
47 \( 1 + (-5.00 - 6.89i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-5.36 - 7.38i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.0544 + 0.167i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.98 + 6.09i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (0.0490 - 0.0675i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (9.83 - 7.14i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-11.4 + 3.72i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-4.01 + 2.91i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-5.50 + 7.57i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + (-0.00380 - 0.0117i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (4.47 + 6.16i)T + (-29.9 + 92.2i)T^{2} \)
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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

−11.03016025558803799431760351705, −10.43474991909362948497716453973, −8.954720231944576896469709044394, −8.256049591273216588803548022576, −7.27828166867594337451524752368, −6.15788361631724251597792850885, −5.01497071856453791226085225028, −4.00466861833423510743680895383, −2.75902133653441714479095564431, −0.43785967200838328837802042496, 2.20766168442599851180752825121, 4.04232467278130427201092784745, 4.87211299049489043187024676072, 5.71808264410957310563075410449, 6.69107922279817790524058684173, 8.158625318607814973445912119552, 9.148889785349272005469536951389, 9.937099731690116433039508024541, 10.55417227808890748948160952306, 12.00198298692300591206560503124

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