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Properties

Label 2-325-65.44-c2-0-2
Degree $2$
Conductor $325$
Sign $0.841 + 0.540i$
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  + (−2.50 − 2.50i)2-s − 5.52i·3-s + 8.55i·4-s + (−13.8 + 13.8i)6-s + (−7.45 + 7.45i)7-s + (11.4 − 11.4i)8-s − 21.5·9-s + (−0.306 − 0.306i)11-s + 47.2·12-s + (−11.9 − 5.08i)13-s + 37.3·14-s − 22.9·16-s + 15.5·17-s + (53.9 + 53.9i)18-s + (3.69 − 3.69i)19-s + ⋯
L(s)  = 1  + (−1.25 − 1.25i)2-s − 1.84i·3-s + 2.13i·4-s + (−2.30 + 2.30i)6-s + (−1.06 + 1.06i)7-s + (1.42 − 1.42i)8-s − 2.39·9-s + (−0.0278 − 0.0278i)11-s + 3.93·12-s + (−0.920 − 0.391i)13-s + 2.66·14-s − 1.43·16-s + 0.912·17-s + (2.99 + 2.99i)18-s + (0.194 − 0.194i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 + 0.540i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.841 + 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.841 + 0.540i$
Analytic conductor: \(8.85560\)
Root analytic conductor: \(2.97583\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (174, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1),\ 0.841 + 0.540i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.264272 - 0.0776416i\)
\(L(\frac12)\) \(\approx\) \(0.264272 - 0.0776416i\)
\(L(2)\) 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 \( 1 + (11.9 + 5.08i)T \)
good2 \( 1 + (2.50 + 2.50i)T + 4iT^{2} \)
3 \( 1 + 5.52iT - 9T^{2} \)
7 \( 1 + (7.45 - 7.45i)T - 49iT^{2} \)
11 \( 1 + (0.306 + 0.306i)T + 121iT^{2} \)
17 \( 1 - 15.5T + 289T^{2} \)
19 \( 1 + (-3.69 + 3.69i)T - 361iT^{2} \)
23 \( 1 + 5.58T + 529T^{2} \)
29 \( 1 - 32.4T + 841T^{2} \)
31 \( 1 + (-13.2 + 13.2i)T - 961iT^{2} \)
37 \( 1 + (23.8 - 23.8i)T - 1.36e3iT^{2} \)
41 \( 1 + (29.7 - 29.7i)T - 1.68e3iT^{2} \)
43 \( 1 - 64.2T + 1.84e3T^{2} \)
47 \( 1 + (20.7 - 20.7i)T - 2.20e3iT^{2} \)
53 \( 1 - 3.22iT - 2.80e3T^{2} \)
59 \( 1 + (-37.8 - 37.8i)T + 3.48e3iT^{2} \)
61 \( 1 + 50.9T + 3.72e3T^{2} \)
67 \( 1 + (-39.1 - 39.1i)T + 4.48e3iT^{2} \)
71 \( 1 + (86.8 - 86.8i)T - 5.04e3iT^{2} \)
73 \( 1 + (54.0 - 54.0i)T - 5.32e3iT^{2} \)
79 \( 1 + 41.3T + 6.24e3T^{2} \)
83 \( 1 + (-49.3 - 49.3i)T + 6.88e3iT^{2} \)
89 \( 1 + (-44.8 - 44.8i)T + 7.92e3iT^{2} \)
97 \( 1 + (-12.6 - 12.6i)T + 9.40e3iT^{2} \)
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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

−11.69952458922345187001578361661, −10.28560029350602463787571295774, −9.434769558487202075447114898961, −8.499215845938327078530655108074, −7.79305227906512251963161764017, −6.82495135898286706381490465295, −5.71578499581493628377968210445, −2.98432175349431097542522563405, −2.50334794783245624218282383764, −1.09148835001438034424193333216, 0.23012029328985454317439387683, 3.36628809081512256788857388115, 4.61957659095596136048640888457, 5.68093534379080434461762768563, 6.75840884127097108522102947318, 7.76859132714231337596332924551, 8.915023722465846458499661425458, 9.609522197240662161109691507552, 10.22735270177223515689919571464, 10.58207265362344091285912123373

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