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Properties

Label 2-325-65.28-c1-0-3
Degree $2$
Conductor $325$
Sign $-0.927 - 0.372i$
Analytic cond. $2.59513$
Root an. cond. $1.61094$
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  + (0.237 − 0.137i)2-s + (−0.611 + 2.28i)3-s + (−0.962 + 1.66i)4-s + (0.168 + 0.627i)6-s + (0.193 − 0.334i)7-s + 1.07i·8-s + (−2.23 − 1.29i)9-s + (−1.12 + 4.21i)11-s + (−3.21 − 3.21i)12-s + (1.35 − 3.34i)13-s − 0.106i·14-s + (−1.77 − 3.07i)16-s + (−1.90 + 0.510i)17-s − 0.710·18-s + (−4.83 + 1.29i)19-s + ⋯
L(s)  = 1  + (0.168 − 0.0971i)2-s + (−0.353 + 1.31i)3-s + (−0.481 + 0.833i)4-s + (0.0686 + 0.256i)6-s + (0.0729 − 0.126i)7-s + 0.381i·8-s + (−0.745 − 0.430i)9-s + (−0.340 + 1.27i)11-s + (−0.928 − 0.928i)12-s + (0.376 − 0.926i)13-s − 0.0283i·14-s + (−0.444 − 0.769i)16-s + (−0.462 + 0.123i)17-s − 0.167·18-s + (−1.11 + 0.297i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 - 0.372i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.927 - 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.927 - 0.372i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ -0.927 - 0.372i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.174577 + 0.903284i\)
\(L(\frac12)\) \(\approx\) \(0.174577 + 0.903284i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 + (-1.35 + 3.34i)T \)
good2 \( 1 + (-0.237 + 0.137i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (0.611 - 2.28i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (-0.193 + 0.334i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.12 - 4.21i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.90 - 0.510i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (4.83 - 1.29i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.322 + 0.0863i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-7.07 + 4.08i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.54 - 2.54i)T - 31iT^{2} \)
37 \( 1 + (-2.41 - 4.17i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.49 - 1.20i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-1.76 - 6.58i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 - 9.83T + 47T^{2} \)
53 \( 1 + (-7.17 - 7.17i)T + 53iT^{2} \)
59 \( 1 + (0.628 + 2.34i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (5.32 - 9.22i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.52 - 3.18i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.12 + 4.20i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + 6.08iT - 73T^{2} \)
79 \( 1 - 3.34iT - 79T^{2} \)
83 \( 1 + 5.18T + 83T^{2} \)
89 \( 1 + (-4.82 - 1.29i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (12.7 + 7.37i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.08070947527771142558772907015, −10.85170873089597201991361219125, −10.28860196103518862269788140344, −9.338656979996404745838327580506, −8.386712228873981697087812759405, −7.39682415053572146463620466273, −5.85080191083535955782100662754, −4.54718080535258238633267872858, −4.24211859634846489143539592356, −2.77873767613548881016257811190, 0.66010466384117537750997516506, 2.14169261217772619308301621978, 4.14784950750830160671807647144, 5.54206122624092203478954006658, 6.29007589268430674359971332973, 7.05337963634418625066641231842, 8.440821371364172925168739460959, 9.074212858185328126557578185661, 10.54960949620150359430893522508, 11.21354644506161349046604199532

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