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Properties

Label 2-325-13.11-c2-0-26
Degree $2$
Conductor $325$
Sign $0.533 + 0.846i$
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  + (0.5 − 1.86i)2-s + (1.36 + 2.36i)3-s + (0.232 + 0.133i)4-s + (5.09 − 1.36i)6-s + (−2.26 − 8.46i)7-s + (5.83 − 5.83i)8-s + (0.767 − 1.33i)9-s + (6.19 + 1.66i)11-s + 0.732i·12-s + (6.5 + 11.2i)13-s − 16.9·14-s + (−7.42 − 12.8i)16-s + (−9.99 − 5.76i)17-s + (−2.09 − 2.09i)18-s + (3.36 − 0.901i)19-s + ⋯
L(s)  = 1  + (0.250 − 0.933i)2-s + (0.455 + 0.788i)3-s + (0.0580 + 0.0334i)4-s + (0.849 − 0.227i)6-s + (−0.323 − 1.20i)7-s + (0.728 − 0.728i)8-s + (0.0853 − 0.147i)9-s + (0.563 + 0.150i)11-s + 0.0610i·12-s + (0.5 + 0.866i)13-s − 1.20·14-s + (−0.464 − 0.804i)16-s + (−0.587 − 0.339i)17-s + (−0.116 − 0.116i)18-s + (0.177 − 0.0474i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.533 + 0.846i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.533 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.533 + 0.846i$
Analytic conductor: \(8.85560\)
Root analytic conductor: \(2.97583\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1),\ 0.533 + 0.846i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.22524 - 1.22808i\)
\(L(\frac12)\) \(\approx\) \(2.22524 - 1.22808i\)
\(L(2)\) 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 \( 1 + (-6.5 - 11.2i)T \)
good2 \( 1 + (-0.5 + 1.86i)T + (-3.46 - 2i)T^{2} \)
3 \( 1 + (-1.36 - 2.36i)T + (-4.5 + 7.79i)T^{2} \)
7 \( 1 + (2.26 + 8.46i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (-6.19 - 1.66i)T + (104. + 60.5i)T^{2} \)
17 \( 1 + (9.99 + 5.76i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-3.36 + 0.901i)T + (312. - 180.5i)T^{2} \)
23 \( 1 + (-8.49 + 4.90i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-5.69 - 9.86i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-1.92 - 1.92i)T + 961iT^{2} \)
37 \( 1 + (-42.1 - 11.2i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (-5.08 + 18.9i)T + (-1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (45 + 25.9i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (0.320 - 0.320i)T - 2.20e3iT^{2} \)
53 \( 1 + 78.7T + 2.80e3T^{2} \)
59 \( 1 + (-10.9 - 40.9i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (49.1 - 85.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (19.9 - 74.5i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (31.0 - 8.31i)T + (4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (48.2 - 48.2i)T - 5.32e3iT^{2} \)
79 \( 1 + 82.7T + 6.24e3T^{2} \)
83 \( 1 + (-69.5 - 69.5i)T + 6.88e3iT^{2} \)
89 \( 1 + (31.8 + 8.52i)T + (6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (74.8 - 20.0i)T + (8.14e3 - 4.70e3i)T^{2} \)
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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

−11.17178872112358897518082412442, −10.36300293611112348210817740666, −9.668088199479296240302795761479, −8.784223765379603878735688364843, −7.24007918227982743996448991467, −6.58735385193861189354131326799, −4.45337570390845562523542902829, −3.98292316368484198281885215113, −2.95753673417211117740540915106, −1.25113196580713985896642917563, 1.70037229089411101578732805789, 2.94020117039647099637247728668, 4.82400216553094199645230349346, 6.00087288129670190534517513287, 6.53396719100565809309101680877, 7.74345265141308813004466692216, 8.328128865549614220583998150414, 9.353404306339669243369457286008, 10.72346518078105438013977404130, 11.64692673658777977613441042504

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