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Properties

Label 2-605-55.9-c1-0-37
Degree $2$
Conductor $605$
Sign $0.615 + 0.788i$
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  + (1.83 + 0.596i)2-s + (−0.304 − 0.418i)3-s + (1.40 + 1.01i)4-s + (−1.88 − 1.21i)5-s + (−0.309 − 0.951i)6-s + (0.526 − 0.725i)7-s + (−0.304 − 0.418i)8-s + (0.844 − 2.59i)9-s + (−2.73 − 3.34i)10-s − 0.896i·12-s + (4.03 + 1.31i)13-s + (1.40 − 1.01i)14-s + (0.0652 + 1.15i)15-s + (−1.37 − 4.24i)16-s + (−0.984 + 0.319i)17-s + (3.10 − 4.26i)18-s + ⋯
L(s)  = 1  + (1.29 + 0.422i)2-s + (−0.175 − 0.241i)3-s + (0.700 + 0.509i)4-s + (−0.840 − 0.541i)5-s + (−0.126 − 0.388i)6-s + (0.199 − 0.274i)7-s + (−0.107 − 0.148i)8-s + (0.281 − 0.866i)9-s + (−0.863 − 1.05i)10-s − 0.258i·12-s + (1.11 + 0.363i)13-s + (0.374 − 0.272i)14-s + (0.0168 + 0.298i)15-s + (−0.344 − 1.06i)16-s + (−0.238 + 0.0775i)17-s + (0.731 − 1.00i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 + 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.615 + 0.788i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.615 + 0.788i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.08771 - 1.01888i\)
\(L(\frac12)\) \(\approx\) \(2.08771 - 1.01888i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.88 + 1.21i)T \)
11 \( 1 \)
good2 \( 1 + (-1.83 - 0.596i)T + (1.61 + 1.17i)T^{2} \)
3 \( 1 + (0.304 + 0.418i)T + (-0.927 + 2.85i)T^{2} \)
7 \( 1 + (-0.526 + 0.725i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (-4.03 - 1.31i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.984 - 0.319i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-5.01 + 3.64i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 6.31iT - 23T^{2} \)
29 \( 1 + (5.60 + 4.07i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.62 - 5.01i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (3.93 - 5.41i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-1.40 + 1.01i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 10.6iT - 43T^{2} \)
47 \( 1 + (-2.41 - 3.32i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-11.3 - 3.69i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (3.82 + 2.78i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.55 + 7.86i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 14.9iT - 67T^{2} \)
71 \( 1 + (-0.678 - 2.08i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-2.87 + 3.96i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.68 + 5.19i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-9.41 + 3.05i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 - 6.46T + 89T^{2} \)
97 \( 1 + (0.624 + 0.202i)T + (78.4 + 57.0i)T^{2} \)
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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

−10.94427882441207673773794295187, −9.478536666718701233857112565723, −8.719183817250321768876680720878, −7.52392482164961813333773790968, −6.75517566781960620857568607020, −5.93042751964358667564503310274, −4.79840430954998626891830831016, −4.09157458585711805890532828110, −3.22699145875864234625970480723, −0.945898997790053654127687062827, 2.04023954911975265092074805484, 3.51245818614825957346546831934, 3.89317294813209442706152370859, 5.26475108666275402840636782919, 5.71743705009923516977817147788, 7.17741229774607467262466705918, 7.968919814040207556571894315663, 9.025153415653962190223674588118, 10.42042105475290226138042833374, 11.03275825598595688588285719812

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