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Properties

Label 2-605-55.14-c1-0-24
Degree $2$
Conductor $605$
Sign $0.275 + 0.961i$
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  + (0.304 + 0.418i)2-s + (−1.83 + 0.596i)3-s + (0.535 − 1.64i)4-s + (−2.23 + 0.126i)5-s + (−0.809 − 0.587i)6-s + (3.18 + 1.03i)7-s + (1.83 − 0.596i)8-s + (0.592 − 0.430i)9-s + (−0.732 − 0.896i)10-s + 3.34i·12-s + (−2.49 − 3.43i)13-s + (0.535 + 1.64i)14-s + (4.02 − 1.56i)15-s + (−1.99 − 1.44i)16-s + (−2.27 + 3.12i)17-s + (0.360 + 0.117i)18-s + ⋯
L(s)  = 1  + (0.215 + 0.296i)2-s + (−1.06 + 0.344i)3-s + (0.267 − 0.823i)4-s + (−0.998 + 0.0563i)5-s + (−0.330 − 0.239i)6-s + (1.20 + 0.390i)7-s + (0.649 − 0.211i)8-s + (0.197 − 0.143i)9-s + (−0.231 − 0.283i)10-s + 0.965i·12-s + (−0.691 − 0.951i)13-s + (0.143 + 0.440i)14-s + (1.03 − 0.403i)15-s + (−0.498 − 0.362i)16-s + (−0.550 + 0.758i)17-s + (0.0849 + 0.0276i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.275 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.275 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.275 + 0.961i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.275 + 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.675272 - 0.508966i\)
\(L(\frac12)\) \(\approx\) \(0.675272 - 0.508966i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.23 - 0.126i)T \)
11 \( 1 \)
good2 \( 1 + (-0.304 - 0.418i)T + (-0.618 + 1.90i)T^{2} \)
3 \( 1 + (1.83 - 0.596i)T + (2.42 - 1.76i)T^{2} \)
7 \( 1 + (-3.18 - 1.03i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (2.49 + 3.43i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (2.27 - 3.12i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.29 + 3.99i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 3.48iT - 23T^{2} \)
29 \( 1 + (-2.14 + 6.58i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-7.06 + 5.13i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.70 - 0.554i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.535 + 1.64i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 6.45iT - 43T^{2} \)
47 \( 1 + (10.8 - 3.53i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.27 - 1.75i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.391 + 1.20i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (9.49 + 6.89i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 2.20iT - 67T^{2} \)
71 \( 1 + (-6.63 - 4.81i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (4.65 + 1.51i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (1.18 - 0.860i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (5.81 - 8.00i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + 0.464T + 89T^{2} \)
97 \( 1 + (-5.37 - 7.39i)T + (-29.9 + 92.2i)T^{2} \)
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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

−10.71390315500004578014599178211, −9.968089682971955695346289117910, −8.472292621749331182107691945765, −7.81966835178726261083177637674, −6.64375200343613725721230724501, −5.82401864147808184457726017972, −4.78096817414379482239033239327, −4.52709062078076962543832959559, −2.41863088303055322893295386791, −0.51941874847623339006759060749, 1.51810312562529421293761052586, 3.18271604010889202114414377171, 4.51229403740428706585796110398, 4.90110657121225410231484989858, 6.56396397061490168515006211795, 7.25761833700559581051682022712, 7.998298101404030835939424434828, 8.852752069291726055696280910134, 10.43483127485178632881292640754, 11.29640948517851618906895722009

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