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Properties

Label 2-605-55.2-c1-0-26
Degree $2$
Conductor $605$
Sign $0.732 - 0.680i$
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  + (1.01 + 1.99i)2-s + (0.221 − 1.39i)3-s + (−1.76 + 2.42i)4-s + (−1.56 − 1.59i)5-s + (3.00 − 0.977i)6-s + (−2.20 − 0.349i)8-s + (0.951 + 0.309i)9-s + (1.58 − 4.74i)10-s + (3 + 3.00i)12-s + (3.98 − 2.03i)13-s + (−2.57 + 1.83i)15-s + (0.309 + 0.951i)16-s + (3.98 + 2.03i)17-s + (0.349 + 2.20i)18-s + (5.11 − 3.71i)19-s + (6.63 − 0.999i)20-s + ⋯
L(s)  = 1  + (0.717 + 1.40i)2-s + (0.127 − 0.806i)3-s + (−0.881 + 1.21i)4-s + (−0.701 − 0.712i)5-s + (1.22 − 0.398i)6-s + (−0.780 − 0.123i)8-s + (0.317 + 0.103i)9-s + (0.500 − 1.50i)10-s + (0.866 + 0.866i)12-s + (1.10 − 0.563i)13-s + (−0.664 + 0.474i)15-s + (0.0772 + 0.237i)16-s + (0.966 + 0.492i)17-s + (0.0824 + 0.520i)18-s + (1.17 − 0.852i)19-s + (1.48 − 0.223i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.732 - 0.680i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.732 - 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.732 - 0.680i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.732 - 0.680i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.01897 + 0.793349i\)
\(L(\frac12)\) \(\approx\) \(2.01897 + 0.793349i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.56 + 1.59i)T \)
11 \( 1 \)
good2 \( 1 + (-1.01 - 1.99i)T + (-1.17 + 1.61i)T^{2} \)
3 \( 1 + (-0.221 + 1.39i)T + (-2.85 - 0.927i)T^{2} \)
7 \( 1 + (6.65 - 2.16i)T^{2} \)
13 \( 1 + (-3.98 + 2.03i)T + (7.64 - 10.5i)T^{2} \)
17 \( 1 + (-3.98 - 2.03i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (-5.11 + 3.71i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (1 - i)T - 23iT^{2} \)
29 \( 1 + (5.11 + 3.71i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.618 + 1.90i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.663 + 4.19i)T + (-35.1 + 11.4i)T^{2} \)
41 \( 1 + (-3.71 - 5.11i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 + (4.19 + 0.663i)T + (44.6 + 14.5i)T^{2} \)
53 \( 1 + (-0.642 - 1.26i)T + (-31.1 + 42.8i)T^{2} \)
59 \( 1 + (3.52 - 4.85i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (6.01 - 1.95i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + (-3 - 3i)T + 67iT^{2} \)
71 \( 1 + (2.47 + 7.60i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-0.699 - 4.41i)T + (-69.4 + 22.5i)T^{2} \)
79 \( 1 + (-1.95 + 6.01i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (4.06 - 7.96i)T + (-48.7 - 67.1i)T^{2} \)
89 \( 1 + 6iT - 89T^{2} \)
97 \( 1 + (8.82 - 4.49i)T + (57.0 - 78.4i)T^{2} \)
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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

−10.96586659815604453630988009742, −9.560711953235669778859793727568, −8.434949204087262050628234583262, −7.76147628037863270900212559328, −7.37167831081458582937755753342, −6.20809053523205177006737974958, −5.45986644303417406663130285378, −4.42123819095937875019990459219, −3.44397057209347989840218322813, −1.24494067323204033816207888665, 1.46720516397248240133572799730, 3.21466027389171016907612218339, 3.57414638173756216716342035370, 4.47804933988266888649128199375, 5.54566217545065887396544596007, 6.94022544939590419703651741616, 7.992811779867769687182888508431, 9.329086107968678333180498735206, 9.982741303522085483250820101289, 10.71993770402097346221914464845

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