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Properties

Label 2-605-55.18-c1-0-21
Degree $2$
Conductor $605$
Sign $0.997 - 0.0692i$
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  + (−0.0900 − 0.568i)2-s + (−0.685 + 0.349i)3-s + (1.58 − 0.515i)4-s + (0.628 + 2.14i)5-s + (0.260 + 0.358i)6-s + (0.541 − 1.06i)7-s + (−0.959 − 1.88i)8-s + (−1.41 + 1.94i)9-s + (1.16 − 0.550i)10-s + (−0.908 + 0.908i)12-s + (6.00 − 0.950i)13-s + (−0.653 − 0.212i)14-s + (−1.18 − 1.25i)15-s + (1.71 − 1.24i)16-s + (−0.366 − 0.0580i)17-s + (1.23 + 0.629i)18-s + ⋯
L(s)  = 1  + (−0.0636 − 0.402i)2-s + (−0.395 + 0.201i)3-s + (0.793 − 0.257i)4-s + (0.281 + 0.959i)5-s + (0.106 + 0.146i)6-s + (0.204 − 0.401i)7-s + (−0.339 − 0.665i)8-s + (−0.471 + 0.649i)9-s + (0.368 − 0.174i)10-s + (−0.262 + 0.262i)12-s + (1.66 − 0.263i)13-s + (−0.174 − 0.0567i)14-s + (−0.304 − 0.323i)15-s + (0.428 − 0.311i)16-s + (−0.0888 − 0.0140i)17-s + (0.291 + 0.148i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0692i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0692i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.997 - 0.0692i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.997 - 0.0692i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65779 + 0.0574704i\)
\(L(\frac12)\) \(\approx\) \(1.65779 + 0.0574704i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.628 - 2.14i)T \)
11 \( 1 \)
good2 \( 1 + (0.0900 + 0.568i)T + (-1.90 + 0.618i)T^{2} \)
3 \( 1 + (0.685 - 0.349i)T + (1.76 - 2.42i)T^{2} \)
7 \( 1 + (-0.541 + 1.06i)T + (-4.11 - 5.66i)T^{2} \)
13 \( 1 + (-6.00 + 0.950i)T + (12.3 - 4.01i)T^{2} \)
17 \( 1 + (0.366 + 0.0580i)T + (16.1 + 5.25i)T^{2} \)
19 \( 1 + (0.425 - 1.30i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-3.48 - 3.48i)T + 23iT^{2} \)
29 \( 1 + (-1.89 - 5.82i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.00 - 1.45i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.65 - 1.35i)T + (21.7 + 29.9i)T^{2} \)
41 \( 1 + (5.82 + 1.89i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (-6.75 + 6.75i)T - 43iT^{2} \)
47 \( 1 + (0.579 + 1.13i)T + (-27.6 + 38.0i)T^{2} \)
53 \( 1 + (-0.0638 - 0.402i)T + (-50.4 + 16.3i)T^{2} \)
59 \( 1 + (-0.178 + 0.0580i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-0.401 - 0.552i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + (-7.14 + 7.14i)T - 67iT^{2} \)
71 \( 1 + (0.836 - 0.607i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (4.07 + 2.07i)T + (42.9 + 59.0i)T^{2} \)
79 \( 1 + (9.41 + 6.83i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-1.78 + 11.2i)T + (-78.9 - 25.6i)T^{2} \)
89 \( 1 + 8.04iT - 89T^{2} \)
97 \( 1 + (8.33 - 1.32i)T + (92.2 - 29.9i)T^{2} \)
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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

−10.64055601880279188152917318173, −10.38606419250076950883069291262, −9.069447411994744245396086270245, −7.908243196043296201264299510470, −6.94542082216133477066847752626, −6.14698751008908913867530620138, −5.36576643992999954689008561048, −3.73112310763286504842160035847, −2.79626105589089084625966651587, −1.44933089410867870130503970639, 1.17818708318842518485018416777, 2.65953056085903356855150226042, 4.11496581361829376904768800869, 5.49034366243108022824088807384, 6.12651080050375292605888793251, 6.81920783705900064295898645475, 8.281112283712131312877087702494, 8.569742518582936989828991136659, 9.580628395587832754537078377786, 10.98656737376642418619098194180

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