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Properties

Label 2-605-11.9-c1-0-9
Degree $2$
Conductor $605$
Sign $0.0219 - 0.999i$
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  + (−0.725 + 2.23i)2-s + (−2.31 + 1.68i)3-s + (−2.84 − 2.06i)4-s + (−0.309 − 0.951i)5-s + (−2.08 − 6.40i)6-s + (−2.73 − 1.98i)7-s + (2.88 − 2.09i)8-s + (1.61 − 4.95i)9-s + 2.34·10-s + 10.0·12-s + (−0.458 + 1.41i)13-s + (6.42 − 4.67i)14-s + (2.31 + 1.68i)15-s + (0.413 + 1.27i)16-s + (1.15 + 3.54i)17-s + (9.91 + 7.20i)18-s + ⋯
L(s)  = 1  + (−0.513 + 1.57i)2-s + (−1.33 + 0.972i)3-s + (−1.42 − 1.03i)4-s + (−0.138 − 0.425i)5-s + (−0.849 − 2.61i)6-s + (−1.03 − 0.751i)7-s + (1.01 − 0.740i)8-s + (0.537 − 1.65i)9-s + 0.742·10-s + 2.91·12-s + (−0.127 + 0.391i)13-s + (1.71 − 1.24i)14-s + (0.598 + 0.434i)15-s + (0.103 + 0.317i)16-s + (0.279 + 0.860i)17-s + (2.33 + 1.69i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0219 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0219 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.0219 - 0.999i$
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: \((2,\ 605,\ (\ :1/2),\ 0.0219 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.259508 + 0.253864i\)
\(L(\frac12)\) \(\approx\) \(0.259508 + 0.253864i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.725 - 2.23i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (2.31 - 1.68i)T + (0.927 - 2.85i)T^{2} \)
7 \( 1 + (2.73 + 1.98i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (0.458 - 1.41i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.15 - 3.54i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (2.60 - 1.88i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 2.51T + 23T^{2} \)
29 \( 1 + (-3.80 - 2.76i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.84 + 8.76i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-2.18 - 1.58i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-3.68 + 2.67i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 5.38T + 43T^{2} \)
47 \( 1 + (-8.51 + 6.18i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.777 - 2.39i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (9.63 + 7.00i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-4.22 - 13.0i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 7.83T + 67T^{2} \)
71 \( 1 + (-0.777 - 2.39i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (2.39 + 1.74i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (2.09 - 6.43i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (2.33 + 7.18i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 10.8T + 89T^{2} \)
97 \( 1 + (-3.40 + 10.4i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.37727775359109747581817656186, −10.00142706307252483043090151766, −9.159411000703475955436674178296, −8.147317950547056908871987256178, −7.06185931213339859467280497733, −6.22327965906253130921141763352, −5.76707882955403948478946785879, −4.60507483927624315834176530951, −3.90764272100315371054923908868, −0.45859017431773310325624719116, 0.75629095680058615664235929600, 2.29321481546453371944435435474, 3.18604077487082636048936129588, 4.78341929640402196187524115898, 6.04023186263050756752048354507, 6.70208961794957827189339202132, 7.85985797397261258191572613989, 9.017355307987993698784014003009, 9.929565611163815194336035444160, 10.63361407621971519272473934089

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