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Properties

Label 2-605-11.5-c1-0-3
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: $\chi_{605} (511, \cdot )$
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

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

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