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Properties

Label 2-605-55.32-c1-0-27
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  + (1.38 − 1.38i)2-s + (−2.07 + 2.07i)3-s − 1.84i·4-s + (−2.03 − 0.933i)5-s + 5.75i·6-s + (1.48 − 1.48i)7-s + (0.212 + 0.212i)8-s − 5.61i·9-s + (−4.11 + 1.52i)10-s + (3.83 + 3.83i)12-s + (1.72 + 1.72i)13-s − 4.12i·14-s + (6.15 − 2.27i)15-s + 4.28·16-s + (3.55 − 3.55i)17-s + (−7.78 − 7.78i)18-s + ⋯
L(s)  = 1  + (0.980 − 0.980i)2-s + (−1.19 + 1.19i)3-s − 0.923i·4-s + (−0.908 − 0.417i)5-s + 2.35i·6-s + (0.561 − 0.561i)7-s + (0.0750 + 0.0750i)8-s − 1.87i·9-s + (−1.30 + 0.481i)10-s + (1.10 + 1.10i)12-s + (0.477 + 0.477i)13-s − 1.10i·14-s + (1.58 − 0.588i)15-s + 1.07·16-s + (0.861 − 0.861i)17-s + (−1.83 − 1.83i)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: $\chi_{605} (362, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.622 + 0.782i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43489 - 0.691815i\)
\(L(\frac12)\) \(\approx\) \(1.43489 - 0.691815i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.03 + 0.933i)T \)
11 \( 1 \)
good2 \( 1 + (-1.38 + 1.38i)T - 2iT^{2} \)
3 \( 1 + (2.07 - 2.07i)T - 3iT^{2} \)
7 \( 1 + (-1.48 + 1.48i)T - 7iT^{2} \)
13 \( 1 + (-1.72 - 1.72i)T + 13iT^{2} \)
17 \( 1 + (-3.55 + 3.55i)T - 17iT^{2} \)
19 \( 1 + 0.415T + 19T^{2} \)
23 \( 1 + (-4.64 + 4.64i)T - 23iT^{2} \)
29 \( 1 - 8.19T + 29T^{2} \)
31 \( 1 + 1.41T + 31T^{2} \)
37 \( 1 + (-0.431 - 0.431i)T + 37iT^{2} \)
41 \( 1 - 0.329iT - 41T^{2} \)
43 \( 1 + (3.73 + 3.73i)T + 43iT^{2} \)
47 \( 1 + (6.46 + 6.46i)T + 47iT^{2} \)
53 \( 1 + (2.30 - 2.30i)T - 53iT^{2} \)
59 \( 1 - 3.04iT - 59T^{2} \)
61 \( 1 - 1.43iT - 61T^{2} \)
67 \( 1 + (4.17 + 4.17i)T + 67iT^{2} \)
71 \( 1 - 7.59T + 71T^{2} \)
73 \( 1 + (5.52 + 5.52i)T + 73iT^{2} \)
79 \( 1 - 3.59T + 79T^{2} \)
83 \( 1 + (1.78 + 1.78i)T + 83iT^{2} \)
89 \( 1 - 7.76iT - 89T^{2} \)
97 \( 1 + (-9.70 - 9.70i)T + 97iT^{2} \)
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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

−10.74272564882502800516857631429, −10.28669794148567440091229636195, −9.065603662182040838669084117437, −7.924635334388883027102775681613, −6.62353102137168532571665893357, −5.23583041588581184417028585998, −4.75888343517502450493667541234, −4.10643164308798389609126381420, −3.19627594142506270861612309691, −0.962125355968218074130757121645, 1.26957639673263409539964319562, 3.30117967761616286616155832209, 4.70016328449199741715809548909, 5.48902953731211864497309592253, 6.23944613243983120562343248576, 6.94834734122985511237800049608, 7.78775455111922051563092441761, 8.310561071290929688070051240547, 10.28180130038283606672329024031, 11.19308024028300204036131833491

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