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Properties

Label 2-605-55.49-c1-0-44
Degree $2$
Conductor $605$
Sign $-0.263 + 0.964i$
Analytic cond. $4.83094$
Root an. cond. $2.19794$
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  + (1.92 − 0.625i)2-s + (1.71 − 2.35i)3-s + (1.69 − 1.22i)4-s + (−1.42 − 1.72i)5-s + (1.82 − 5.60i)6-s + (1.88 + 2.59i)7-s + (0.109 − 0.150i)8-s + (−1.69 − 5.22i)9-s + (−3.81 − 2.43i)10-s − 6.09i·12-s + (0.617 − 0.200i)13-s + (5.25 + 3.81i)14-s + (−6.50 + 0.395i)15-s + (−1.17 + 3.62i)16-s + (−1.13 − 0.367i)17-s + (−6.53 − 8.99i)18-s + ⋯
L(s)  = 1  + (1.36 − 0.441i)2-s + (0.989 − 1.36i)3-s + (0.846 − 0.614i)4-s + (−0.635 − 0.771i)5-s + (0.743 − 2.28i)6-s + (0.713 + 0.981i)7-s + (0.0385 − 0.0530i)8-s + (−0.566 − 1.74i)9-s + (−1.20 − 0.768i)10-s − 1.76i·12-s + (0.171 − 0.0556i)13-s + (1.40 + 1.02i)14-s + (−1.67 + 0.102i)15-s + (−0.294 + 0.905i)16-s + (−0.274 − 0.0890i)17-s + (−1.54 − 2.11i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.263 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.263 + 0.964i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.263 + 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.25206 - 2.94910i\)
\(L(\frac12)\) \(\approx\) \(2.25206 - 2.94910i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.42 + 1.72i)T \)
11 \( 1 \)
good2 \( 1 + (-1.92 + 0.625i)T + (1.61 - 1.17i)T^{2} \)
3 \( 1 + (-1.71 + 2.35i)T + (-0.927 - 2.85i)T^{2} \)
7 \( 1 + (-1.88 - 2.59i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (-0.617 + 0.200i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.13 + 0.367i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.52 - 1.11i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 4.35iT - 23T^{2} \)
29 \( 1 + (3.36 - 2.44i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.44 - 7.51i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-1.20 - 1.66i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-2.67 - 1.94i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 10.9iT - 43T^{2} \)
47 \( 1 + (-1.71 + 2.35i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (6.10 - 1.98i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (3.00 - 2.18i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (2.40 - 7.40i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 2.37iT - 67T^{2} \)
71 \( 1 + (-4.85 + 14.9i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (3.35 + 4.62i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-4.71 - 14.5i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-10.1 - 3.28i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 + (14.3 - 4.67i)T + (78.4 - 57.0i)T^{2} \)
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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

−10.90997049662330835726150142862, −9.029062582197799278674664755272, −8.569097372478582459150076385674, −7.84200055161137270832038604051, −6.73410428836554058746645405054, −5.60900582978936744842713847854, −4.72710899182144336237235976772, −3.52439448253003850468177556677, −2.52318478265324944865419013674, −1.52631089131845144592627371094, 2.71031534699503214064365685679, 3.70845108730713554038532985452, 4.17354433620178532898506400496, 4.93279384324218685578861201313, 6.21282510206212913181128772526, 7.48000301940352606779034446507, 7.936905481667251687429709724944, 9.305367799298310890690214992053, 10.04187433014946149532470139962, 11.12408470633144273965448334122

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