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Properties

Label 2-605-11.5-c1-0-22
Degree $2$
Conductor $605$
Sign $-0.836 - 0.548i$
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  + (−0.848 − 2.61i)2-s + (0.170 + 0.123i)3-s + (−4.47 + 3.25i)4-s + (−0.309 + 0.951i)5-s + (0.178 − 0.550i)6-s + (1.87 − 1.36i)7-s + (7.84 + 5.70i)8-s + (−0.913 − 2.81i)9-s + 2.74·10-s − 1.16·12-s + (−0.165 − 0.507i)13-s + (−5.15 − 3.74i)14-s + (−0.170 + 0.123i)15-s + (4.80 − 14.7i)16-s + (0.748 − 2.30i)17-s + (−6.56 + 4.76i)18-s + ⋯
L(s)  = 1  + (−0.599 − 1.84i)2-s + (0.0984 + 0.0715i)3-s + (−2.23 + 1.62i)4-s + (−0.138 + 0.425i)5-s + (0.0729 − 0.224i)6-s + (0.710 − 0.516i)7-s + (2.77 + 2.01i)8-s + (−0.304 − 0.936i)9-s + 0.867·10-s − 0.336·12-s + (−0.0457 − 0.140i)13-s + (−1.37 − 1.00i)14-s + (−0.0440 + 0.0319i)15-s + (1.20 − 3.69i)16-s + (0.181 − 0.558i)17-s + (−1.54 + 1.12i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 - 0.548i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.836 - 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.836 - 0.548i$
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: \((2,\ 605,\ (\ :1/2),\ -0.836 - 0.548i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.189682 + 0.634744i\)
\(L(\frac12)\) \(\approx\) \(0.189682 + 0.634744i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.848 + 2.61i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.170 - 0.123i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + (-1.87 + 1.36i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (0.165 + 0.507i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.748 + 2.30i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (4.00 + 2.91i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 4.53T + 23T^{2} \)
29 \( 1 + (-4.44 + 3.22i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.322 - 0.993i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (6.05 - 4.40i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (8.57 + 6.23i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 4.32T + 43T^{2} \)
47 \( 1 + (5.47 + 3.97i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.40 + 4.31i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-5.21 + 3.78i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.10 + 6.46i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 0.721T + 67T^{2} \)
71 \( 1 + (-1.40 + 4.31i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (0.864 - 0.627i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.43 - 4.41i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-4.20 + 12.9i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 + (-1.47 - 4.52i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.29422285409023421936944569161, −9.555624768515359992151735002141, −8.595865387619684276245922976797, −8.003844403871308508619072963380, −6.75102163495851456661626216755, −4.96515084636563631604613359643, −3.99352073707510832087444260943, −3.14177420652642272106532453360, −1.97698376250206777232245643291, −0.45913000009866961977144739185, 1.70695026731536871990286491749, 4.21330244151022650724880938831, 5.10747759884034253741372600734, 5.79736850299307972402786457083, 6.77867472172661653320904808811, 7.954118856973407744610405022163, 8.242257535633936544519400353162, 8.909666690974204559913874890029, 10.02783133284587557089400610348, 10.77281759417557001193011648896

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