↑

Properties

Label 2-325-65.63-c1-0-11
Degree $2$
Conductor $325$
Sign $0.488 + 0.872i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.457 − 0.791i)2-s + (0.0872 − 0.325i)3-s + (0.582 − 1.00i)4-s + (−0.297 + 0.0797i)6-s + (3.61 + 2.08i)7-s − 2.89·8-s + (2.49 + 1.44i)9-s + (5.01 + 1.34i)11-s + (−0.277 − 0.277i)12-s + (−2.87 − 2.17i)13-s − 3.81i·14-s + (0.157 + 0.272i)16-s + (−2.91 + 0.780i)17-s − 2.63i·18-s + (−1.43 − 5.35i)19-s + ⋯
L(s)  = 1  + (−0.323 − 0.559i)2-s + (0.0503 − 0.188i)3-s + (0.291 − 0.504i)4-s + (−0.121 + 0.0325i)6-s + (1.36 + 0.788i)7-s − 1.02·8-s + (0.833 + 0.481i)9-s + (1.51 + 0.404i)11-s + (−0.0801 − 0.0801i)12-s + (−0.798 − 0.602i)13-s − 1.01i·14-s + (0.0393 + 0.0682i)16-s + (−0.706 + 0.189i)17-s − 0.621i·18-s + (−0.329 − 1.22i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.488 + 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.488 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.488 + 0.872i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ 0.488 + 0.872i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23396 - 0.723641i\)
\(L(\frac12)\) \(\approx\) \(1.23396 - 0.723641i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.87 + 2.17i)T \)
good2 \( 1 + (0.457 + 0.791i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.0872 + 0.325i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (-3.61 - 2.08i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.01 - 1.34i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (2.91 - 0.780i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.43 + 5.35i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.65 - 0.442i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (7.08 - 4.08i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.64 + 2.64i)T + 31iT^{2} \)
37 \( 1 + (-6.72 + 3.88i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.228 - 0.853i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (0.224 + 0.839i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 - 1.44iT - 47T^{2} \)
53 \( 1 + (0.405 + 0.405i)T + 53iT^{2} \)
59 \( 1 + (9.44 - 2.53i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.28 - 3.95i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.55 + 6.15i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.65 - 0.712i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 - 6.02T + 73T^{2} \)
79 \( 1 - 12.5iT - 79T^{2} \)
83 \( 1 - 8.44iT - 83T^{2} \)
89 \( 1 + (-3.79 + 14.1i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (0.386 - 0.670i)T + (-48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.27987711218850776538620895989, −10.80236047828087126925634818025, −9.471629680742514833166710427599, −8.976675614271949878733527986523, −7.64466048747948557247827574411, −6.69480556065831482547864449023, −5.39466436926339623099042901718, −4.40815467712189625806858671363, −2.38757420956209066159752159952, −1.51035926540518483891774211751, 1.67882104057708973735871122894, 3.76223958858053679933492080454, 4.50912723347981004060557134840, 6.23557281092280817208567536606, 7.10814462903259873861351713749, 7.82537573073501963542502940752, 8.880039057422195690845565093864, 9.658259888459736127934040080788, 11.00008976301315959112729455906, 11.68834572070194414509167152824

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