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Properties

Label 2-325-65.59-c2-0-23
Degree $2$
Conductor $325$
Sign $0.983 - 0.179i$
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  + (−0.206 − 0.772i)2-s + (5.13 + 2.96i)3-s + (2.91 − 1.68i)4-s + (1.22 − 4.57i)6-s + (0.592 − 2.21i)7-s + (−4.16 − 4.16i)8-s + (13.0 + 22.6i)9-s + (1.95 + 7.30i)11-s + 19.9·12-s + (−9.61 + 8.74i)13-s − 1.82·14-s + (4.36 − 7.56i)16-s + (−1.74 − 3.02i)17-s + (14.7 − 14.7i)18-s + (6.49 − 24.2i)19-s + ⋯
L(s)  = 1  + (−0.103 − 0.386i)2-s + (1.71 + 0.987i)3-s + (0.727 − 0.420i)4-s + (0.204 − 0.763i)6-s + (0.0846 − 0.315i)7-s + (−0.520 − 0.520i)8-s + (1.45 + 2.51i)9-s + (0.177 + 0.663i)11-s + 1.65·12-s + (−0.739 + 0.673i)13-s − 0.130·14-s + (0.272 − 0.472i)16-s + (−0.102 − 0.177i)17-s + (0.820 − 0.820i)18-s + (0.341 − 1.27i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.179i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.983 - 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.983 - 0.179i$
Analytic conductor: \(8.85560\)
Root analytic conductor: \(2.97583\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1),\ 0.983 - 0.179i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.15609 + 0.285752i\)
\(L(\frac12)\) \(\approx\) \(3.15609 + 0.285752i\)
\(L(2)\) 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 \( 1 + (9.61 - 8.74i)T \)
good2 \( 1 + (0.206 + 0.772i)T + (-3.46 + 2i)T^{2} \)
3 \( 1 + (-5.13 - 2.96i)T + (4.5 + 7.79i)T^{2} \)
7 \( 1 + (-0.592 + 2.21i)T + (-42.4 - 24.5i)T^{2} \)
11 \( 1 + (-1.95 - 7.30i)T + (-104. + 60.5i)T^{2} \)
17 \( 1 + (1.74 + 3.02i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-6.49 + 24.2i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (-1.63 + 2.82i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-1.52 + 2.64i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (25.6 + 25.6i)T + 961iT^{2} \)
37 \( 1 + (32.1 - 8.61i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (-35.6 + 9.55i)T + (1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (25.1 + 43.4i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (2.51 + 2.51i)T + 2.20e3iT^{2} \)
53 \( 1 - 52.2iT - 2.80e3T^{2} \)
59 \( 1 + (20.3 + 5.46i)T + (3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (21.6 + 37.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-6.36 - 23.7i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (-17.6 + 66.0i)T + (-4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (40.6 + 40.6i)T + 5.32e3iT^{2} \)
79 \( 1 + 27.1T + 6.24e3T^{2} \)
83 \( 1 + (41.5 - 41.5i)T - 6.88e3iT^{2} \)
89 \( 1 + (-33.5 - 125. i)T + (-6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (-55.8 - 14.9i)T + (8.14e3 + 4.70e3i)T^{2} \)
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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

−11.07339342227175265052478670834, −10.33319433199414134490850204591, −9.463115116339130057980689374131, −9.056810551366676469606375404508, −7.60071423674072898604275789010, −6.99432636290635645207431550222, −5.02945889790982997659407435525, −4.02689972122724738380007646073, −2.79397424911262241505527488032, −1.94415853945749434167117915968, 1.65058467779169914699221785961, 2.81500414571846130328874634237, 3.59946787193351496604481389375, 5.82204700217616800273619382030, 6.90430846883741640011985625738, 7.66673582839713360903462103509, 8.311683686232549086235479490081, 9.038423267131603238131117480511, 10.21468363364647101209205852508, 11.70994980839240080332497819629

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