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Properties

Label 2-325-13.4-c1-0-3
Degree $2$
Conductor $325$
Sign $0.943 - 0.331i$
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  + (−1.18 − 0.682i)2-s + (0.199 − 0.346i)3-s + (−0.0676 − 0.117i)4-s + (−0.472 + 0.273i)6-s + (−3.15 + 1.82i)7-s + 2.91i·8-s + (1.42 + 2.45i)9-s + (2.83 + 1.63i)11-s − 0.0541·12-s + (3.53 + 0.710i)13-s + 4.97·14-s + (1.85 − 3.21i)16-s + (−1.08 − 1.88i)17-s − 3.87i·18-s + (−0.742 + 0.428i)19-s + ⋯
L(s)  = 1  + (−0.836 − 0.482i)2-s + (0.115 − 0.199i)3-s + (−0.0338 − 0.0586i)4-s + (−0.193 + 0.111i)6-s + (−1.19 + 0.688i)7-s + 1.03i·8-s + (0.473 + 0.819i)9-s + (0.855 + 0.494i)11-s − 0.0156·12-s + (0.980 + 0.196i)13-s + 1.32·14-s + (0.463 − 0.803i)16-s + (−0.263 − 0.456i)17-s − 0.914i·18-s + (−0.170 + 0.0983i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.331i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 - 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.943 - 0.331i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ 0.943 - 0.331i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.725818 + 0.123898i\)
\(L(\frac12)\) \(\approx\) \(0.725818 + 0.123898i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.53 - 0.710i)T \)
good2 \( 1 + (1.18 + 0.682i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-0.199 + 0.346i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (3.15 - 1.82i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.83 - 1.63i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.08 + 1.88i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.742 - 0.428i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.382 - 0.662i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.53 + 2.65i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.41iT - 31T^{2} \)
37 \( 1 + (-9.40 - 5.42i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.59 + 3.80i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.91 - 3.32i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 4.68iT - 47T^{2} \)
53 \( 1 + 3.04T + 53T^{2} \)
59 \( 1 + (11.5 - 6.66i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.51 - 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.02 - 2.32i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-9.38 + 5.41i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 7.68iT - 73T^{2} \)
79 \( 1 + 10.2T + 79T^{2} \)
83 \( 1 + 1.18iT - 83T^{2} \)
89 \( 1 + (-2.82 - 1.63i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.54 - 2.62i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.52193642976074158535785423125, −10.54584621207685455413022558186, −9.711051319966320986828386783246, −9.059877508618045265892459505283, −8.202263363992825692480368021816, −6.87759445990793809318303764949, −5.92349608423060009119452050834, −4.53173992752560056521637839903, −2.86699819202162449906572762809, −1.51384152221418389801922862485, 0.76362274395095017902581965301, 3.52568075888256982321521289538, 4.00587769273546007884389638333, 6.32275417979924101000991805157, 6.60412506598000166244400133121, 7.85509485324477369766997037419, 8.887012832638313185501364243163, 9.512222042471972511784987390344, 10.25458912844279085686814114582, 11.38851754547668509395397681228

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