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Properties

Label 2-375-25.6-c1-0-0
Degree $2$
Conductor $375$
Sign $-0.230 + 0.973i$
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  + (0.830 + 2.55i)2-s + (−0.809 + 0.587i)3-s + (−4.22 + 3.06i)4-s + (−2.17 − 1.57i)6-s − 1.68·7-s + (−7.00 − 5.08i)8-s + (0.309 − 0.951i)9-s + (0.333 + 1.02i)11-s + (1.61 − 4.96i)12-s + (−0.827 + 2.54i)13-s + (−1.40 − 4.31i)14-s + (3.95 − 12.1i)16-s + (−3.18 − 2.31i)17-s + 2.68·18-s + (0.952 + 0.692i)19-s + ⋯
L(s)  = 1  + (0.587 + 1.80i)2-s + (−0.467 + 0.339i)3-s + (−2.11 + 1.53i)4-s + (−0.887 − 0.644i)6-s − 0.637·7-s + (−2.47 − 1.79i)8-s + (0.103 − 0.317i)9-s + (0.100 + 0.309i)11-s + (0.465 − 1.43i)12-s + (−0.229 + 0.706i)13-s + (−0.374 − 1.15i)14-s + (0.989 − 3.04i)16-s + (−0.771 − 0.560i)17-s + 0.633·18-s + (0.218 + 0.158i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.230 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $-0.230 + 0.973i$
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: \((2,\ 375,\ (\ :1/2),\ -0.230 + 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.509416 - 0.644021i\)
\(L(\frac12)\) \(\approx\) \(0.509416 - 0.644021i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.830 - 2.55i)T + (-1.61 + 1.17i)T^{2} \)
7 \( 1 + 1.68T + 7T^{2} \)
11 \( 1 + (-0.333 - 1.02i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (0.827 - 2.54i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (3.18 + 2.31i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.952 - 0.692i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-1.25 - 3.86i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (4.81 - 3.50i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-5.74 - 4.17i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.41 + 4.35i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (3.47 - 10.7i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 2.58T + 43T^{2} \)
47 \( 1 + (1.55 - 1.12i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.06 - 1.49i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.412 + 1.27i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.24 + 6.92i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-10.0 - 7.31i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-4.84 + 3.51i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.01 - 3.13i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-3.23 + 2.35i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-7.17 - 5.21i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-4.77 - 14.6i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-8.71 + 6.32i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.33367353795969574582469549349, −11.30255546828952678267038220627, −9.690699998234524832246600604329, −9.210574098444903775413068999993, −8.024071337580063442746513356347, −6.91775512583343898204349519686, −6.47912682943198732535927204477, −5.30764536457987881032139406716, −4.55480957465812347928117042440, −3.41886538652666908726028435938, 0.48066038901943911894846694573, 2.16144580836999926273144312560, 3.31269693182740360229046748109, 4.45157957691086875193043812923, 5.53453591754150199054568462088, 6.52054292832820330950186200199, 8.238291505442606693002562950241, 9.336505818108578881876363426279, 10.19169493904365242681546498440, 10.87403672840055260217199024713

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