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Properties

Label 2-605-11.9-c1-0-3
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: $\chi_{605} (251, \cdot )$
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

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

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