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Properties

Label 2-605-11.3-c1-0-18
Degree $2$
Conductor $605$
Sign $0.530 - 0.847i$
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  + (−1.40 + 1.01i)2-s + (0.618 + 1.90i)3-s + (0.309 − 0.951i)4-s + (−0.809 − 0.587i)5-s + (−2.80 − 2.03i)6-s + (1.07 − 3.29i)7-s + (−0.535 − 1.64i)8-s + (−0.809 + 0.587i)9-s + 1.73·10-s + 1.99·12-s + (1.85 + 5.70i)14-s + (0.618 − 1.90i)15-s + (4.04 + 2.93i)16-s + (5.60 + 4.07i)17-s + (0.535 − 1.64i)18-s + (−2.14 − 6.58i)19-s + ⋯
L(s)  = 1  + (−0.990 + 0.719i)2-s + (0.356 + 1.09i)3-s + (0.154 − 0.475i)4-s + (−0.361 − 0.262i)5-s + (−1.14 − 0.831i)6-s + (0.404 − 1.24i)7-s + (−0.189 − 0.582i)8-s + (−0.269 + 0.195i)9-s + 0.547·10-s + 0.577·12-s + (0.495 + 1.52i)14-s + (0.159 − 0.491i)15-s + (1.01 + 0.734i)16-s + (1.35 + 0.987i)17-s + (0.126 − 0.388i)18-s + (−0.491 − 1.51i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 - 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.530 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.530 - 0.847i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (366, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.530 - 0.847i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.846999 + 0.469344i\)
\(L(\frac12)\) \(\approx\) \(0.846999 + 0.469344i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 \)
good2 \( 1 + (1.40 - 1.01i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-0.618 - 1.90i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (-1.07 + 3.29i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-5.60 - 4.07i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.14 + 6.58i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (3.23 - 2.35i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-3.09 + 9.51i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.14 - 6.58i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 3.46T + 43T^{2} \)
47 \( 1 + (1.85 + 5.70i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-4.85 + 3.52i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (5.60 + 4.07i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 10T + 67T^{2} \)
71 \( 1 + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.14 - 6.58i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-5.60 + 4.07i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-14.0 - 10.1i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (-8.09 + 5.87i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.55623476987057525370711844144, −9.726202435597066511421424248279, −9.028516647261305525317668515212, −8.253994170637330776388392830036, −7.43925619585050226167169611359, −6.70768343748098254035632719833, −5.15163864048819828599044137134, −4.13325773461897984884421397223, −3.41724132915514042518579894405, −0.918101070317190732550379607724, 1.19232867376134542255304192006, 2.23443187652323205697244099792, 3.14067774808261312500893397121, 5.10938672904927729522846297193, 6.08589075590464709800155581005, 7.41289719940712658135336273060, 8.016377682732352016186127586694, 8.703999490451474897465133574370, 9.577133913798394092562364736321, 10.46614418642154578742650483498

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