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Properties

Label 2-325-13.11-c2-0-17
Degree $2$
Conductor $325$
Sign $0.741 - 0.671i$
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  + (−0.0928 + 0.346i)2-s + (0.295 + 0.511i)3-s + (3.35 + 1.93i)4-s + (−0.204 + 0.0548i)6-s + (0.0877 + 0.327i)7-s + (−1.99 + 1.99i)8-s + (4.32 − 7.49i)9-s + (6.41 + 1.71i)11-s + 2.28i·12-s + (9.96 − 8.34i)13-s − 0.121·14-s + (7.23 + 12.5i)16-s + (25.1 + 14.5i)17-s + (2.19 + 2.19i)18-s + (−27.1 + 7.26i)19-s + ⋯
L(s)  = 1  + (−0.0464 + 0.173i)2-s + (0.0983 + 0.170i)3-s + (0.838 + 0.483i)4-s + (−0.0341 + 0.00913i)6-s + (0.0125 + 0.0467i)7-s + (−0.249 + 0.249i)8-s + (0.480 − 0.832i)9-s + (0.583 + 0.156i)11-s + 0.190i·12-s + (0.766 − 0.642i)13-s − 0.00868·14-s + (0.452 + 0.783i)16-s + (1.48 + 0.854i)17-s + (0.121 + 0.121i)18-s + (−1.42 + 0.382i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.741 - 0.671i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.741 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.741 - 0.671i$
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: \((2,\ 325,\ (\ :1),\ 0.741 - 0.671i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.03935 + 0.786368i\)
\(L(\frac12)\) \(\approx\) \(2.03935 + 0.786368i\)
\(L(2)\) 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 \( 1 + (-9.96 + 8.34i)T \)
good2 \( 1 + (0.0928 - 0.346i)T + (-3.46 - 2i)T^{2} \)
3 \( 1 + (-0.295 - 0.511i)T + (-4.5 + 7.79i)T^{2} \)
7 \( 1 + (-0.0877 - 0.327i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (-6.41 - 1.71i)T + (104. + 60.5i)T^{2} \)
17 \( 1 + (-25.1 - 14.5i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (27.1 - 7.26i)T + (312. - 180.5i)T^{2} \)
23 \( 1 + (10.5 - 6.10i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (19.0 + 33.0i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-27.5 - 27.5i)T + 961iT^{2} \)
37 \( 1 + (-26.6 - 7.13i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (8.23 - 30.7i)T + (-1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (-0.391 - 0.226i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (45.8 - 45.8i)T - 2.20e3iT^{2} \)
53 \( 1 + 17.3T + 2.80e3T^{2} \)
59 \( 1 + (4.05 + 15.1i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-13.8 + 23.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-25.0 + 93.6i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (26.9 - 7.21i)T + (4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (-39.7 + 39.7i)T - 5.32e3iT^{2} \)
79 \( 1 - 3.63T + 6.24e3T^{2} \)
83 \( 1 + (107. + 107. i)T + 6.88e3iT^{2} \)
89 \( 1 + (57.1 + 15.3i)T + (6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (74.2 - 19.9i)T + (8.14e3 - 4.70e3i)T^{2} \)
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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

−11.58853359994334844093147117126, −10.53806196976583697502162984779, −9.741776531739551482704801350709, −8.437591262900654976292272143673, −7.81443600062686660542248850283, −6.48586402023497340467839411614, −5.98245366634203321704073792231, −4.09449500569726164385031933748, −3.23563985476302346561424730826, −1.52206870753546481653379449589, 1.27056843739536461825205130434, 2.48110448105428607386852629892, 4.01830662610310294274851940018, 5.43467140017639195832926555289, 6.50863828796714517284752154369, 7.30962584899439805714081068348, 8.413550070479085176913500999704, 9.595623818917972078802455128488, 10.46269949719912061654344226255, 11.24113962372257251422575135304

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