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Properties

Label 2-375-25.21-c1-0-12
Degree $2$
Conductor $375$
Sign $-0.451 + 0.892i$
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  + (0.726 − 2.23i)2-s + (0.809 + 0.587i)3-s + (−2.85 − 2.07i)4-s + (1.90 − 1.38i)6-s + 3.48·7-s + (−2.90 + 2.10i)8-s + (0.309 + 0.951i)9-s + (0.905 − 2.78i)11-s + (−1.08 − 3.35i)12-s + (−0.579 − 1.78i)13-s + (2.52 − 7.78i)14-s + (0.427 + 1.31i)16-s + (−5.48 + 3.98i)17-s + 2.35·18-s + (2.38 − 1.73i)19-s + ⋯
L(s)  = 1  + (0.513 − 1.58i)2-s + (0.467 + 0.339i)3-s + (−1.42 − 1.03i)4-s + (0.776 − 0.564i)6-s + 1.31·7-s + (−1.02 + 0.745i)8-s + (0.103 + 0.317i)9-s + (0.273 − 0.840i)11-s + (−0.314 − 0.968i)12-s + (−0.160 − 0.494i)13-s + (0.676 − 2.08i)14-s + (0.106 + 0.328i)16-s + (−1.33 + 0.967i)17-s + 0.554·18-s + (0.547 − 0.397i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $-0.451 + 0.892i$
Analytic conductor: \(2.99439\)
Root analytic conductor: \(1.73043\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{375} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 375,\ (\ :1/2),\ -0.451 + 0.892i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09544 - 1.78113i\)
\(L(\frac12)\) \(\approx\) \(1.09544 - 1.78113i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 \)
good2 \( 1 + (-0.726 + 2.23i)T + (-1.61 - 1.17i)T^{2} \)
7 \( 1 - 3.48T + 7T^{2} \)
11 \( 1 + (-0.905 + 2.78i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (0.579 + 1.78i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (5.48 - 3.98i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-2.38 + 1.73i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (1.69 - 5.22i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (2.06 + 1.50i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.338 - 0.245i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.61 - 4.98i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.518 - 1.59i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 10.9T + 43T^{2} \)
47 \( 1 + (-6.06 - 4.40i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (3.00 + 2.18i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.19 + 6.76i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (1.98 - 6.12i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (8.12 - 5.90i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (0.589 + 0.428i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.11 - 3.41i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (2.48 + 1.80i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (8.18 - 5.94i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-0.0888 + 0.273i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-8.42 - 6.11i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.16487956509491096716024387160, −10.54482313890802311666216703912, −9.441972378159324122778346242753, −8.625681596591236239677911810366, −7.62551799875520171584724241708, −5.75998757937682948464831681183, −4.66821742038238360200380798934, −3.86173802114174368265136784312, −2.65682233971177324139194721110, −1.45018833184350723055690086518, 2.11798935258072537706157752519, 4.23343386030735785928907075845, 4.80170702999371897426143964936, 6.03406366201810448910930914263, 7.18336989280549453186140994245, 7.54438456709966766653224823663, 8.638952372385771455995106237405, 9.271975555399675696706116753708, 10.85132831425209829866779178490, 11.93180271634348257778856093054

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