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Properties

Label 2-375-25.4-c1-0-14
Degree $2$
Conductor $375$
Sign $-0.476 + 0.879i$
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  + (0.234 + 0.0762i)2-s + (0.587 − 0.809i)3-s + (−1.56 − 1.13i)4-s + (0.199 − 0.145i)6-s − 1.24i·7-s + (−0.571 − 0.786i)8-s + (−0.309 − 0.951i)9-s + (0.794 − 2.44i)11-s + (−1.84 + 0.599i)12-s + (−4.45 + 1.44i)13-s + (0.0950 − 0.292i)14-s + (1.12 + 3.46i)16-s + (−3.43 − 4.72i)17-s − 0.246i·18-s + (3.37 − 2.45i)19-s + ⋯
L(s)  = 1  + (0.165 + 0.0539i)2-s + (0.339 − 0.467i)3-s + (−0.784 − 0.569i)4-s + (0.0814 − 0.0592i)6-s − 0.471i·7-s + (−0.201 − 0.278i)8-s + (−0.103 − 0.317i)9-s + (0.239 − 0.736i)11-s + (−0.532 + 0.172i)12-s + (−1.23 + 0.401i)13-s + (0.0254 − 0.0781i)14-s + (0.281 + 0.865i)16-s + (−0.832 − 1.14i)17-s − 0.0581i·18-s + (0.773 − 0.562i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.476 + 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.476 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $-0.476 + 0.879i$
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: \((2,\ 375,\ (\ :1/2),\ -0.476 + 0.879i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.563852 - 0.946353i\)
\(L(\frac12)\) \(\approx\) \(0.563852 - 0.946353i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.234 - 0.0762i)T + (1.61 + 1.17i)T^{2} \)
7 \( 1 + 1.24iT - 7T^{2} \)
11 \( 1 + (-0.794 + 2.44i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (4.45 - 1.44i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (3.43 + 4.72i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-3.37 + 2.45i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (1.52 + 0.496i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (2.60 + 1.89i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-7.43 + 5.40i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.21 + 0.394i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.68 - 8.27i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 3.88iT - 43T^{2} \)
47 \( 1 + (1.88 - 2.59i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-7.83 + 10.7i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.97 - 6.07i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.18 + 3.63i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-5.95 - 8.19i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (11.2 + 8.15i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-10.8 - 3.51i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-9.29 - 6.74i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (1.66 + 2.29i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (-0.426 + 1.31i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-0.0179 + 0.0246i)T + (-29.9 - 92.2i)T^{2} \)
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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

−11.21091461301184245715753478706, −9.821050806309071557664792752672, −9.408743439651781462175221221430, −8.317459833915131268226624200095, −7.26301636805327216227893433333, −6.31758948288033205408088481873, −5.08036373806156446268665475384, −4.14802719754420234148988687058, −2.61468663713236108061234183451, −0.69228056857897590057504093316, 2.38633117538768607821373527915, 3.69105210757453743389836131422, 4.64824028325109602177411353855, 5.57991527065984536694042202161, 7.15650270376072483529028141948, 8.124776580884820062804013262948, 8.952739776113231462858262115979, 9.744233259708250034346013764819, 10.55939243117686065709291917452, 12.09604310748361660607293581269

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