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Properties

Label 2-605-11.9-c1-0-4
Degree $2$
Conductor $605$
Sign $-0.955 - 0.296i$
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  + (−0.535 + 1.64i)2-s + (0.809 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (−0.309 − 0.951i)5-s + (0.535 + 1.64i)6-s + (−1.40 − 1.01i)7-s + (−1.40 + 1.01i)8-s + (−0.618 + 1.90i)9-s + 1.73·10-s − 0.999·12-s + (−1.07 + 3.29i)13-s + (2.42 − 1.76i)14-s + (−0.809 − 0.587i)15-s + (−1.54 − 4.75i)16-s + (2.14 + 6.58i)17-s + (−2.80 − 2.03i)18-s + ⋯
L(s)  = 1  + (−0.378 + 1.16i)2-s + (0.467 − 0.339i)3-s + (−0.404 − 0.293i)4-s + (−0.138 − 0.425i)5-s + (0.218 + 0.672i)6-s + (−0.529 − 0.384i)7-s + (−0.495 + 0.359i)8-s + (−0.206 + 0.634i)9-s + 0.547·10-s − 0.288·12-s + (−0.296 + 0.913i)13-s + (0.648 − 0.471i)14-s + (−0.208 − 0.151i)15-s + (−0.386 − 1.18i)16-s + (0.519 + 1.59i)17-s + (−0.660 − 0.479i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 - 0.296i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.955 - 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.955 - 0.296i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.955 - 0.296i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.138750 + 0.914648i\)
\(L(\frac12)\) \(\approx\) \(0.138750 + 0.914648i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 \)
good2 \( 1 + (0.535 - 1.64i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (-0.809 + 0.587i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 + (1.40 + 1.01i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (1.07 - 3.29i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-2.14 - 6.58i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (2.80 - 2.03i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (2.47 - 7.60i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-6.47 - 4.70i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-9.80 + 7.12i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 8.66T + 43T^{2} \)
47 \( 1 + (7.28 - 5.29i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-1.85 + 5.70i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-9.70 - 7.05i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.67 + 8.23i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 5T + 67T^{2} \)
71 \( 1 + (3.70 + 11.4i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-3.21 + 9.88i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.07 + 3.29i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 + (3.09 - 9.51i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.90416980350494891759329441644, −9.921421949312531924598188066706, −8.874926902039091740169993560281, −8.293010066092016285170188126556, −7.59253363957056207641923362941, −6.71302769358317850691961709327, −5.92761041343036030485253318068, −4.75475205834497974420977867429, −3.42704816867098118272470018467, −1.89915896000177776016912566591, 0.52383441525499122481647027165, 2.60544502620474250636658627943, 2.99549793055167692081639477413, 4.12447565200165754889916359143, 5.69243658969422356189073197161, 6.66196701887455674268692707894, 7.82651881624910283145853854626, 8.959233980662818203370266249788, 9.634478515283333399975695751512, 10.05048518035841728065744749558

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