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Properties

Label 2-325-65.58-c1-0-5
Degree $2$
Conductor $325$
Sign $-0.507 - 0.861i$
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  + (1.98 + 1.14i)2-s + (−1.80 + 0.483i)3-s + (1.62 + 2.82i)4-s + (−4.14 − 1.10i)6-s + (1.60 + 2.78i)7-s + 2.88i·8-s + (0.430 − 0.248i)9-s + (−3.45 + 0.925i)11-s + (−4.30 − 4.30i)12-s + (3.07 + 1.87i)13-s + 7.37i·14-s + (−0.0520 + 0.0901i)16-s + (−1.55 + 5.80i)17-s + 1.13·18-s + (2.03 − 7.60i)19-s + ⋯
L(s)  = 1  + (1.40 + 0.810i)2-s + (−1.04 + 0.279i)3-s + (0.814 + 1.41i)4-s + (−1.69 − 0.453i)6-s + (0.607 + 1.05i)7-s + 1.02i·8-s + (0.143 − 0.0827i)9-s + (−1.04 + 0.279i)11-s + (−1.24 − 1.24i)12-s + (0.853 + 0.520i)13-s + 1.97i·14-s + (−0.0130 + 0.0225i)16-s + (−0.376 + 1.40i)17-s + 0.268·18-s + (0.467 − 1.74i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.507 - 0.861i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.507 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.507 - 0.861i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (318, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ -0.507 - 0.861i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.976845 + 1.70837i\)
\(L(\frac12)\) \(\approx\) \(0.976845 + 1.70837i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.07 - 1.87i)T \)
good2 \( 1 + (-1.98 - 1.14i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (1.80 - 0.483i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (-1.60 - 2.78i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.45 - 0.925i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.55 - 5.80i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-2.03 + 7.60i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.296 + 1.10i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-0.877 - 0.506i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.47 + 4.47i)T - 31iT^{2} \)
37 \( 1 + (-0.372 + 0.645i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.65 - 6.16i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (1.25 + 0.335i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 - 8.22T + 47T^{2} \)
53 \( 1 + (3.01 + 3.01i)T + 53iT^{2} \)
59 \( 1 + (-6.45 - 1.72i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.928 - 1.60i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.23 + 4.17i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (9.01 + 2.41i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + 7.93iT - 73T^{2} \)
79 \( 1 + 10.1iT - 79T^{2} \)
83 \( 1 + 9.01T + 83T^{2} \)
89 \( 1 + (-0.176 - 0.657i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (11.7 - 6.80i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.93693353228119297049344577512, −11.38046185390878573729876636229, −10.46477027702814074671075907533, −8.867003888246850131080897732168, −7.86232192048740112486384495533, −6.52615604660527885286679114490, −5.85970820696923203347633386440, −5.06127318434415731834996075960, −4.35184877878072989892781004089, −2.62416131962745063571804271269, 1.13417157938198563209953370160, 2.98838382934730281520383605101, 4.20075608598791522499848096993, 5.29292998660401928782710531129, 5.83255811514215769640181787360, 7.13061874227932383321476245384, 8.272014272643172747813890824349, 10.20857261023960906286361481881, 10.75747461682353290670296465720, 11.45629399171549908113566611242

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