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Properties

Label 2-605-55.32-c1-0-32
Degree $2$
Conductor $605$
Sign $0.762 + 0.647i$
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  + (1.07 − 1.07i)2-s + (0.563 − 0.563i)3-s − 0.310i·4-s + (2.23 + 0.152i)5-s − 1.21i·6-s + (−0.135 + 0.135i)7-s + (1.81 + 1.81i)8-s + 2.36i·9-s + (2.56 − 2.23i)10-s + (−0.175 − 0.175i)12-s + (−2.18 − 2.18i)13-s + 0.291i·14-s + (1.34 − 1.17i)15-s + 4.52·16-s + (2.62 − 2.62i)17-s + (2.54 + 2.54i)18-s + ⋯
L(s)  = 1  + (0.760 − 0.760i)2-s + (0.325 − 0.325i)3-s − 0.155i·4-s + (0.997 + 0.0684i)5-s − 0.494i·6-s + (−0.0511 + 0.0511i)7-s + (0.642 + 0.642i)8-s + 0.788i·9-s + (0.810 − 0.706i)10-s + (−0.0505 − 0.0505i)12-s + (−0.607 − 0.607i)13-s + 0.0777i·14-s + (0.346 − 0.302i)15-s + 1.13·16-s + (0.635 − 0.635i)17-s + (0.599 + 0.599i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 + 0.647i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.762 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.762 + 0.647i$
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: \((2,\ 605,\ (\ :1/2),\ 0.762 + 0.647i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.67874 - 0.983568i\)
\(L(\frac12)\) \(\approx\) \(2.67874 - 0.983568i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.23 - 0.152i)T \)
11 \( 1 \)
good2 \( 1 + (-1.07 + 1.07i)T - 2iT^{2} \)
3 \( 1 + (-0.563 + 0.563i)T - 3iT^{2} \)
7 \( 1 + (0.135 - 0.135i)T - 7iT^{2} \)
13 \( 1 + (2.18 + 2.18i)T + 13iT^{2} \)
17 \( 1 + (-2.62 + 2.62i)T - 17iT^{2} \)
19 \( 1 - 0.743T + 19T^{2} \)
23 \( 1 + (1.14 - 1.14i)T - 23iT^{2} \)
29 \( 1 + 9.54T + 29T^{2} \)
31 \( 1 - 0.350T + 31T^{2} \)
37 \( 1 + (3.82 + 3.82i)T + 37iT^{2} \)
41 \( 1 + 6.69iT - 41T^{2} \)
43 \( 1 + (3.72 + 3.72i)T + 43iT^{2} \)
47 \( 1 + (8.74 + 8.74i)T + 47iT^{2} \)
53 \( 1 + (-6.38 + 6.38i)T - 53iT^{2} \)
59 \( 1 - 9.62iT - 59T^{2} \)
61 \( 1 + 5.89iT - 61T^{2} \)
67 \( 1 + (-4.13 - 4.13i)T + 67iT^{2} \)
71 \( 1 + 11.4T + 71T^{2} \)
73 \( 1 + (-1.69 - 1.69i)T + 73iT^{2} \)
79 \( 1 + 0.670T + 79T^{2} \)
83 \( 1 + (-11.7 - 11.7i)T + 83iT^{2} \)
89 \( 1 + 7.92iT - 89T^{2} \)
97 \( 1 + (-0.975 - 0.975i)T + 97iT^{2} \)
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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

−10.59346020996189240119964684685, −9.965318411189705813371227824646, −8.886413562068384914540531917620, −7.80115601250672417994313173521, −7.11209051266663542567614065476, −5.49091066854935199571016261594, −5.14546465441723793187909153040, −3.62771634855408283076837877224, −2.58845836890010483011826883170, −1.80294410330417237687505090192, 1.60456537564154736258821260169, 3.27061239762725706618503973794, 4.36935191899631955275368958862, 5.32126869974424486089268134562, 6.19142667211516109991733333084, 6.81859132927576117358047250969, 7.959595411223694028220986049210, 9.223228360294442515546904741742, 9.758625775312931156812667742888, 10.47038211336765690038395603281

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