Properties

Label 2-375-25.19-c1-0-10
Degree $2$
Conductor $375$
Sign $-0.646 + 0.762i$
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  + (−2.32 + 0.754i)2-s + (−0.587 − 0.809i)3-s + (3.19 − 2.32i)4-s + (1.97 + 1.43i)6-s − 3.44i·7-s + (−2.80 + 3.85i)8-s + (−0.309 + 0.951i)9-s + (−1.00 − 3.10i)11-s + (−3.76 − 1.22i)12-s + (3.07 + 0.998i)13-s + (2.59 + 7.98i)14-s + (1.15 − 3.54i)16-s + (−2.97 + 4.08i)17-s − 2.44i·18-s + (−2.49 − 1.81i)19-s + ⋯
L(s)  = 1  + (−1.64 + 0.533i)2-s + (−0.339 − 0.467i)3-s + (1.59 − 1.16i)4-s + (0.805 + 0.585i)6-s − 1.30i·7-s + (−0.991 + 1.36i)8-s + (−0.103 + 0.317i)9-s + (−0.304 − 0.936i)11-s + (−1.08 − 0.352i)12-s + (0.852 + 0.277i)13-s + (0.693 + 2.13i)14-s + (0.288 − 0.887i)16-s + (−0.720 + 0.991i)17-s − 0.575i·18-s + (−0.571 − 0.415i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.646 + 0.762i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.646 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $-0.646 + 0.762i$
Analytic conductor: \(2.99439\)
Root analytic conductor: \(1.73043\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{375} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 375,\ (\ :1/2),\ -0.646 + 0.762i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.129314 - 0.279306i\)
\(L(\frac12)\) \(\approx\) \(0.129314 - 0.279306i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.32 - 0.754i)T + (1.61 - 1.17i)T^{2} \)
7 \( 1 + 3.44iT - 7T^{2} \)
11 \( 1 + (1.00 + 3.10i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-3.07 - 0.998i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (2.97 - 4.08i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (2.49 + 1.81i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-1.47 + 0.478i)T + (18.6 - 13.5i)T^{2} \)
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 \( 1 + (5.47 + 1.77i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.67 + 5.15i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 2.53iT - 43T^{2} \)
47 \( 1 + (4.15 + 5.72i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (5.96 + 8.21i)T + (-16.3 + 50.4i)T^{2} \)
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 \( 1 + (1.08 - 1.49i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (-0.577 + 0.419i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.78 - 0.581i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (10.7 - 7.83i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (2.32 - 3.20i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + (2.63 + 8.11i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-6.26 - 8.61i)T + (-29.9 + 92.2i)T^{2} \)
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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

−10.91639498158191479704100514428, −10.21077063464351090033932209851, −8.899191801878916970091504461861, −8.349196787619787456753052021281, −7.31008645722434478569675453879, −6.68472072247106203962199017105, −5.74295857746142835325650133653, −3.89165308472777589433646660009, −1.73871719883103201366781079634, −0.36047364483587687684371299629, 1.84556192377696583715251594112, 3.05387952417620905069298510409, 4.86959620890789528109663859661, 6.14823895143341788223609198774, 7.31162471220192745802170576596, 8.407650390224868037419334069994, 9.115510274563967875275266500450, 9.724827990424295000896796406964, 10.73386779114193423273147920787, 11.34542299855305310472019051975

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