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Properties

Label 2-375-25.6-c1-0-3
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: $\chi_{375} (151, \cdot )$
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

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

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