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Properties

Label 2-325-5.4-c5-0-8
Degree $2$
Conductor $325$
Sign $0.447 + 0.894i$
Analytic cond. $52.1247$
Root an. cond. $7.21974$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.20i·2-s + 29.5i·3-s + 21.7·4-s − 94.8·6-s − 11.5i·7-s + 172. i·8-s − 632.·9-s − 596.·11-s + 642. i·12-s − 169i·13-s + 37.1·14-s + 142.·16-s + 2.09e3i·17-s − 2.02e3i·18-s − 35.4·19-s + ⋯
L(s)  = 1  + 0.566i·2-s + 1.89i·3-s + 0.678·4-s − 1.07·6-s − 0.0892i·7-s + 0.951i·8-s − 2.60·9-s − 1.48·11-s + 1.28i·12-s − 0.277i·13-s + 0.0506·14-s + 0.139·16-s + 1.75i·17-s − 1.47i·18-s − 0.0225·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(52.1247\)
Root analytic conductor: \(7.21974\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :5/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.7659285357\)
\(L(\frac12)\) \(\approx\) \(0.7659285357\)
\(L(\frac{7}{2})\) 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 \( 1 + 169iT \)
good2 \( 1 - 3.20iT - 32T^{2} \)
3 \( 1 - 29.5iT - 243T^{2} \)
7 \( 1 + 11.5iT - 1.68e4T^{2} \)
11 \( 1 + 596.T + 1.61e5T^{2} \)
17 \( 1 - 2.09e3iT - 1.41e6T^{2} \)
19 \( 1 + 35.4T + 2.47e6T^{2} \)
23 \( 1 + 2.78e3iT - 6.43e6T^{2} \)
29 \( 1 - 370.T + 2.05e7T^{2} \)
31 \( 1 - 5.05e3T + 2.86e7T^{2} \)
37 \( 1 - 4.12e3iT - 6.93e7T^{2} \)
41 \( 1 + 1.81e4T + 1.15e8T^{2} \)
43 \( 1 - 7.90e3iT - 1.47e8T^{2} \)
47 \( 1 + 1.31e4iT - 2.29e8T^{2} \)
53 \( 1 + 3.82e4iT - 4.18e8T^{2} \)
59 \( 1 + 1.78e4T + 7.14e8T^{2} \)
61 \( 1 - 7.39e3T + 8.44e8T^{2} \)
67 \( 1 + 2.33e4iT - 1.35e9T^{2} \)
71 \( 1 + 3.31e4T + 1.80e9T^{2} \)
73 \( 1 - 1.08e4iT - 2.07e9T^{2} \)
79 \( 1 - 1.77e4T + 3.07e9T^{2} \)
83 \( 1 - 8.42e4iT - 3.93e9T^{2} \)
89 \( 1 - 4.64e4T + 5.58e9T^{2} \)
97 \( 1 - 1.54e5iT - 8.58e9T^{2} \)
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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

−11.10012890585024235559798136618, −10.43600209282465574285677905962, −10.06799090164732486957934654880, −8.453174926825239788858410364217, −8.162413412389258628832969670734, −6.46913417080508007443984357826, −5.51719684595395905630295848273, −4.75152295609782881031287881609, −3.47725811711752918898264119339, −2.43072010846673202855284819410, 0.18475467836405768476338680412, 1.28991080725586359721882999750, 2.41218186200037421504477115783, 2.98992942803779082318067691672, 5.27461904287277902549120057686, 6.28396249907723201603327899088, 7.36955737541825208652762611246, 7.59707640601827919319344262269, 8.931060139785972302820022867597, 10.26464985074492341994357727996

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