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Properties

Label 2-375-25.9-c1-0-12
Degree $2$
Conductor $375$
Sign $0.390 + 0.920i$
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  + (0.903 − 1.24i)2-s + (0.951 − 0.309i)3-s + (−0.112 − 0.346i)4-s + (0.475 − 1.46i)6-s − 1.68i·7-s + (2.39 + 0.777i)8-s + (0.809 − 0.587i)9-s + (2.40 + 1.75i)11-s + (−0.214 − 0.294i)12-s + (−0.136 − 0.188i)13-s + (−2.09 − 1.52i)14-s + (3.71 − 2.70i)16-s + (−7.09 − 2.30i)17-s − 1.53i·18-s + (−0.232 + 0.716i)19-s + ⋯
L(s)  = 1  + (0.639 − 0.879i)2-s + (0.549 − 0.178i)3-s + (−0.0563 − 0.173i)4-s + (0.193 − 0.597i)6-s − 0.637i·7-s + (0.845 + 0.274i)8-s + (0.269 − 0.195i)9-s + (0.726 + 0.527i)11-s + (−0.0618 − 0.0851i)12-s + (−0.0379 − 0.0522i)13-s + (−0.560 − 0.407i)14-s + (0.929 − 0.675i)16-s + (−1.72 − 0.559i)17-s − 0.362i·18-s + (−0.0534 + 0.164i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.390 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $0.390 + 0.920i$
Analytic conductor: \(2.99439\)
Root analytic conductor: \(1.73043\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{375} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 375,\ (\ :1/2),\ 0.390 + 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.95545 - 1.29529i\)
\(L(\frac12)\) \(\approx\) \(1.95545 - 1.29529i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.951 + 0.309i)T \)
5 \( 1 \)
good2 \( 1 + (-0.903 + 1.24i)T + (-0.618 - 1.90i)T^{2} \)
7 \( 1 + 1.68iT - 7T^{2} \)
11 \( 1 + (-2.40 - 1.75i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.136 + 0.188i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (7.09 + 2.30i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.232 - 0.716i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-0.512 + 0.706i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (2.12 + 6.53i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (3.03 - 9.33i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-5.95 - 8.19i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (3.07 - 2.23i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 5.27iT - 43T^{2} \)
47 \( 1 + (8.14 - 2.64i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (5.68 - 1.84i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (3.11 - 2.26i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-3.55 - 2.58i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (1.70 + 0.554i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (1.35 + 4.16i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (8.84 - 12.1i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.27 + 7.01i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-4.13 - 1.34i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (-9.79 - 7.11i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-9.01 + 2.92i)T + (78.4 - 57.0i)T^{2} \)
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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

−11.37727332831111681552866744511, −10.48960343807500221330568452233, −9.529113613433136410431546259597, −8.476162295104724951206320946754, −7.38345641042057828915101773526, −6.58745364557097729013120879081, −4.75072511498907717144992375409, −4.05759778213931186484258922097, −2.88260975543983214034667639423, −1.66763248784969604333322285776, 2.02602951512193184668373718993, 3.71803949187632933915169238737, 4.69375812478399269208651168807, 5.86625765997406713287300819065, 6.61564370977710051061413579215, 7.66441865967153103297886907416, 8.759689790030162708644165582928, 9.412259204135422207444386898988, 10.74162224087844141981592486298, 11.46681168256737140125757366901

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