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Properties

Label 2-375-25.6-c1-0-6
Degree $2$
Conductor $375$
Sign $-0.951 + 0.308i$
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  + (−0.754 − 2.32i)2-s + (−0.809 + 0.587i)3-s + (−3.19 + 2.32i)4-s + (1.97 + 1.43i)6-s + 3.44·7-s + (3.85 + 2.80i)8-s + (0.309 − 0.951i)9-s + (−1.00 − 3.10i)11-s + (1.22 − 3.76i)12-s + (0.998 − 3.07i)13-s + (−2.59 − 7.98i)14-s + (1.15 − 3.54i)16-s + (−4.08 − 2.97i)17-s − 2.44·18-s + (2.49 + 1.81i)19-s + ⋯
L(s)  = 1  + (−0.533 − 1.64i)2-s + (−0.467 + 0.339i)3-s + (−1.59 + 1.16i)4-s + (0.805 + 0.585i)6-s + 1.30·7-s + (1.36 + 0.991i)8-s + (0.103 − 0.317i)9-s + (−0.304 − 0.936i)11-s + (0.352 − 1.08i)12-s + (0.277 − 0.852i)13-s + (−0.693 − 2.13i)14-s + (0.288 − 0.887i)16-s + (−0.991 − 0.720i)17-s − 0.575·18-s + (0.571 + 0.415i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 + 0.308i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.951 + 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $-0.951 + 0.308i$
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: \((2,\ 375,\ (\ :1/2),\ -0.951 + 0.308i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.122785 - 0.776581i\)
\(L(\frac12)\) \(\approx\) \(0.122785 - 0.776581i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.754 + 2.32i)T + (-1.61 + 1.17i)T^{2} \)
7 \( 1 - 3.44T + 7T^{2} \)
11 \( 1 + (1.00 + 3.10i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-0.998 + 3.07i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (4.08 + 2.97i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.49 - 1.81i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (0.478 + 1.47i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-2.52 + 1.83i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (6.02 + 4.37i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.77 + 5.47i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.67 + 5.15i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 2.53T + 43T^{2} \)
47 \( 1 + (-5.72 + 4.15i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (8.21 - 5.96i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.534 + 1.64i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.42 - 7.45i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (1.49 + 1.08i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-0.577 + 0.419i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.581 - 1.78i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-10.7 + 7.83i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-3.20 - 2.32i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-2.63 - 8.11i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (8.61 - 6.26i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.99072259720016200221430153128, −10.46032761936432033143445327055, −9.322431802452453567976710986333, −8.532164045079021955877055458722, −7.64684852159460881852551039784, −5.77370624478958172680785284077, −4.71158126189251017707235978082, −3.59986952673953312358041495062, −2.27538233188217695799508256650, −0.71642549981840436284931786188, 1.67087400574516844321469803076, 4.54088440879824562093802416833, 5.11933241249149869451873345905, 6.33373754613487371077156553922, 7.09344431427846453837093868902, 7.892653497319691289688565136194, 8.679863367570320098269180114100, 9.620466115940727550566203601983, 10.83276727668626010971949640918, 11.63904435636105470228477086839

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