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Properties

Label 2-325-65.28-c1-0-15
Degree $2$
Conductor $325$
Sign $0.573 + 0.819i$
Analytic cond. $2.59513$
Root an. cond. $1.61094$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.29 − 1.32i)2-s + (−0.335 + 1.25i)3-s + (2.51 − 4.34i)4-s + (0.889 + 3.31i)6-s + (−0.0561 + 0.0972i)7-s − 8.00i·8-s + (1.14 + 0.658i)9-s + (0.479 − 1.78i)11-s + (4.60 + 4.60i)12-s + (−2.96 + 2.05i)13-s + 0.297i·14-s + (−5.58 − 9.67i)16-s + (2.63 − 0.706i)17-s + 3.49·18-s + (−6.72 + 1.80i)19-s + ⋯
L(s)  = 1  + (1.62 − 0.936i)2-s + (−0.193 + 0.723i)3-s + (1.25 − 2.17i)4-s + (0.363 + 1.35i)6-s + (−0.0212 + 0.0367i)7-s − 2.83i·8-s + (0.380 + 0.219i)9-s + (0.144 − 0.539i)11-s + (1.32 + 1.32i)12-s + (−0.821 + 0.569i)13-s + 0.0795i·14-s + (−1.39 − 2.41i)16-s + (0.639 − 0.171i)17-s + 0.823·18-s + (−1.54 + 0.413i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.573 + 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.573 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.573 + 0.819i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ 0.573 + 0.819i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.59448 - 1.35114i\)
\(L(\frac12)\) \(\approx\) \(2.59448 - 1.35114i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.96 - 2.05i)T \)
good2 \( 1 + (-2.29 + 1.32i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (0.335 - 1.25i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (0.0561 - 0.0972i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.479 + 1.78i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-2.63 + 0.706i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (6.72 - 1.80i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.10 - 0.831i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (4.03 - 2.32i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.624 - 0.624i)T - 31iT^{2} \)
37 \( 1 + (-0.737 - 1.27i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.24 + 1.40i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (1.00 + 3.76i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 - 0.345T + 47T^{2} \)
53 \( 1 + (-3.59 - 3.59i)T + 53iT^{2} \)
59 \( 1 + (-0.332 - 1.24i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.39 + 2.41i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.124 - 0.0721i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.41 + 5.28i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + 9.06iT - 73T^{2} \)
79 \( 1 + 15.1iT - 79T^{2} \)
83 \( 1 + 8.53T + 83T^{2} \)
89 \( 1 + (-0.549 - 0.147i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-12.9 - 7.48i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.57085315578815194888651281516, −10.70160877178665107816625731301, −10.16690695808122627077795109313, −9.086024401267612100278392796827, −7.24733371307160906345677144813, −6.07183217465086733493048799063, −5.08525280134262425069293461876, −4.30980954427213889450494985272, −3.34460376520069752825403061805, −1.90585133946080632581520320306, 2.34342666056408033567308632419, 3.83064283062783203623128888183, 4.84372533243697722965582293548, 5.89158232817749341783011476407, 6.84235858952263748203845962344, 7.37089060854196002483894344660, 8.386442346570276706045343941892, 9.994760441520722488572183242200, 11.37766508490514828178797402865, 12.26212916516387536333423550960

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