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Properties

Label 2-325-5.4-c5-0-71
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  + 6.07i·2-s − 25.6i·3-s − 4.94·4-s + 155.·6-s + 137. i·7-s + 164. i·8-s − 413.·9-s + 169.·11-s + 126. i·12-s − 169i·13-s − 834.·14-s − 1.15e3·16-s − 487. i·17-s − 2.51e3i·18-s − 2.38e3·19-s + ⋯
L(s)  = 1  + 1.07i·2-s − 1.64i·3-s − 0.154·4-s + 1.76·6-s + 1.05i·7-s + 0.908i·8-s − 1.70·9-s + 0.421·11-s + 0.254i·12-s − 0.277i·13-s − 1.13·14-s − 1.13·16-s − 0.408i·17-s − 1.82i·18-s − 1.51·19-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

\(L(3)\) \(\approx\) \(0.8343181178\)
\(L(\frac12)\) \(\approx\) \(0.8343181178\)
\(L(\frac{7}{2})\) 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 \( 1 - 6.07iT - 32T^{2} \)
3 \( 1 + 25.6iT - 243T^{2} \)
7 \( 1 - 137. iT - 1.68e4T^{2} \)
11 \( 1 - 169.T + 1.61e5T^{2} \)
17 \( 1 + 487. iT - 1.41e6T^{2} \)
19 \( 1 + 2.38e3T + 2.47e6T^{2} \)
23 \( 1 + 4.14e3iT - 6.43e6T^{2} \)
29 \( 1 - 4.57e3T + 2.05e7T^{2} \)
31 \( 1 + 2.95e3T + 2.86e7T^{2} \)
37 \( 1 - 5.73e3iT - 6.93e7T^{2} \)
41 \( 1 + 1.70e3T + 1.15e8T^{2} \)
43 \( 1 + 1.09e4iT - 1.47e8T^{2} \)
47 \( 1 + 7.91e3iT - 2.29e8T^{2} \)
53 \( 1 + 2.64e4iT - 4.18e8T^{2} \)
59 \( 1 + 1.12e4T + 7.14e8T^{2} \)
61 \( 1 + 4.02e4T + 8.44e8T^{2} \)
67 \( 1 + 6.48e4iT - 1.35e9T^{2} \)
71 \( 1 + 4.70e4T + 1.80e9T^{2} \)
73 \( 1 + 4.27e4iT - 2.07e9T^{2} \)
79 \( 1 + 9.00e4T + 3.07e9T^{2} \)
83 \( 1 - 1.64e4iT - 3.93e9T^{2} \)
89 \( 1 - 3.56e4T + 5.58e9T^{2} \)
97 \( 1 + 1.21e5iT - 8.58e9T^{2} \)
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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

−10.63629028761202070187271341357, −8.776387954484754840817112145412, −8.439210623721892381742409479095, −7.42874489963799323343533642504, −6.45108318449258411226025877962, −6.18170812831406850587190191681, −4.91778359903567618563532810606, −2.66653389151328845499909775463, −1.87515480593145921068027818920, −0.20320156896442999409901519302, 1.41823243721137462808196812323, 2.99144136971711744263983233692, 4.01848971009343847533116733737, 4.38928529926288370143696484975, 6.00034906093732428536283742014, 7.26961981637941221000800025739, 8.777312801857995170595357600103, 9.621421253475044928351954759514, 10.33886649888557775465129395205, 10.87874385118669052686812728114

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