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Properties

Label 2-325-65.24-c2-0-3
Degree $2$
Conductor $325$
Sign $-0.973 + 0.228i$
Analytic cond. $8.85560$
Root an. cond. $2.97583$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.20 + 0.590i)2-s + (−1.83 + 1.05i)3-s + (1.03 + 0.598i)4-s + (−4.66 + 1.24i)6-s + (−1.17 + 0.314i)7-s + (−4.51 − 4.51i)8-s + (−2.26 + 3.91i)9-s + (−13.3 − 3.57i)11-s − 2.53·12-s + (−3.48 + 12.5i)13-s − 2.76·14-s + (−9.67 − 16.7i)16-s + (4.73 − 8.20i)17-s + (−7.28 + 7.28i)18-s + (−23.4 + 6.28i)19-s + ⋯
L(s)  = 1  + (1.10 + 0.295i)2-s + (−0.610 + 0.352i)3-s + (0.259 + 0.149i)4-s + (−0.776 + 0.208i)6-s + (−0.167 + 0.0449i)7-s + (−0.564 − 0.564i)8-s + (−0.251 + 0.435i)9-s + (−1.21 − 0.324i)11-s − 0.211·12-s + (−0.268 + 0.963i)13-s − 0.197·14-s + (−0.604 − 1.04i)16-s + (0.278 − 0.482i)17-s + (−0.404 + 0.404i)18-s + (−1.23 + 0.330i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.228i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.973 + 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.973 + 0.228i$
Analytic conductor: \(8.85560\)
Root analytic conductor: \(2.97583\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1),\ -0.973 + 0.228i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0391895 - 0.338517i\)
\(L(\frac12)\) \(\approx\) \(0.0391895 - 0.338517i\)
\(L(2)\) 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 \( 1 + (3.48 - 12.5i)T \)
good2 \( 1 + (-2.20 - 0.590i)T + (3.46 + 2i)T^{2} \)
3 \( 1 + (1.83 - 1.05i)T + (4.5 - 7.79i)T^{2} \)
7 \( 1 + (1.17 - 0.314i)T + (42.4 - 24.5i)T^{2} \)
11 \( 1 + (13.3 + 3.57i)T + (104. + 60.5i)T^{2} \)
17 \( 1 + (-4.73 + 8.20i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (23.4 - 6.28i)T + (312. - 180.5i)T^{2} \)
23 \( 1 + (-5.80 - 10.0i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-2.30 - 3.99i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (21.8 + 21.8i)T + 961iT^{2} \)
37 \( 1 + (5.14 - 19.1i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (14.6 - 54.8i)T + (-1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (-35.4 + 61.4i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-22.8 - 22.8i)T + 2.20e3iT^{2} \)
53 \( 1 + 3.82iT - 2.80e3T^{2} \)
59 \( 1 + (-17.8 - 66.5i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-20.0 + 34.7i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-56.1 - 15.0i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (53.0 - 14.2i)T + (4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (-34.0 - 34.0i)T + 5.32e3iT^{2} \)
79 \( 1 - 27.0T + 6.24e3T^{2} \)
83 \( 1 + (-20.0 + 20.0i)T - 6.88e3iT^{2} \)
89 \( 1 + (-30.1 - 8.08i)T + (6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (-46.1 - 172. i)T + (-8.14e3 + 4.70e3i)T^{2} \)
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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

−11.99016742123026611420717453749, −11.11289935871479193578566383713, −10.21399877239382840569425427780, −9.192926768631820506319558721200, −7.927042516977375347281577576059, −6.68067890460345410467280082933, −5.69673347476343096373350664101, −5.01730187698713001015509724610, −4.06366723558735064792842877318, −2.60632725145365255259340819372, 0.11182822587118602272374539634, 2.45669458801467018973589768758, 3.61360292267457516417275752473, 4.95869537244629793647805356341, 5.66277204954140844128726904810, 6.64715672405619626673899377892, 7.941996087260840047178798876492, 8.961069563819511231230094342429, 10.40405339424675165491153400169, 11.04114583025200389639640811381

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