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Properties

Label 2-605-55.18-c1-0-14
Degree $2$
Conductor $605$
Sign $-0.622 - 0.782i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.622 - 0.782i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.622 - 0.782i$
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: \((2,\ 605,\ (\ :1/2),\ -0.622 - 0.782i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.598970 + 1.24196i\)
\(L(\frac12)\) \(\approx\) \(0.598970 + 1.24196i\)
\(L(\frac{3}{2})\) 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

−11.15853029888013989078193635271, −10.08722449452296838495336306789, −9.541987712864798475822994029395, −7.980152403094651759277405558044, −7.29167644067598673124903515561, −6.43658152035927137307051916471, −5.62965641075968971508548607095, −4.80600223095251411673172384723, −2.91431361497254272592559327235, −2.21440519909650958608024971158, 0.74209886974546054865417534510, 2.25483170537617976877554671477, 3.47245739511061806060653047982, 4.78057938325486353695841539896, 5.75349114365228415138314389564, 6.80239251563497786650934579419, 7.51796070613164808221749196544, 8.666345055064451353313076664140, 9.642056861543966056733715439574, 10.37614652314700646664728902544

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