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Properties

Label 2-605-55.32-c1-0-0
Degree $2$
Conductor $605$
Sign $-0.0495 + 0.998i$
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  + (−0.345 + 0.345i)2-s + (−0.805 + 0.805i)3-s + 1.76i·4-s + (−2.18 − 0.490i)5-s − 0.556i·6-s + (−2.06 + 2.06i)7-s + (−1.29 − 1.29i)8-s + 1.70i·9-s + (0.922 − 0.583i)10-s + (−1.41 − 1.41i)12-s + (−2.06 − 2.06i)13-s − 1.42i·14-s + (2.15 − 1.36i)15-s − 2.62·16-s + (3.72 − 3.72i)17-s + (−0.587 − 0.587i)18-s + ⋯
L(s)  = 1  + (−0.244 + 0.244i)2-s + (−0.465 + 0.465i)3-s + 0.880i·4-s + (−0.975 − 0.219i)5-s − 0.227i·6-s + (−0.779 + 0.779i)7-s + (−0.459 − 0.459i)8-s + 0.567i·9-s + (0.291 − 0.184i)10-s + (−0.409 − 0.409i)12-s + (−0.573 − 0.573i)13-s − 0.380i·14-s + (0.555 − 0.351i)15-s − 0.656·16-s + (0.903 − 0.903i)17-s + (−0.138 − 0.138i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0495 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0495 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.0495 + 0.998i$
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: \((2,\ 605,\ (\ :1/2),\ -0.0495 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0232891 - 0.0244724i\)
\(L(\frac12)\) \(\approx\) \(0.0232891 - 0.0244724i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.18 + 0.490i)T \)
11 \( 1 \)
good2 \( 1 + (0.345 - 0.345i)T - 2iT^{2} \)
3 \( 1 + (0.805 - 0.805i)T - 3iT^{2} \)
7 \( 1 + (2.06 - 2.06i)T - 7iT^{2} \)
13 \( 1 + (2.06 + 2.06i)T + 13iT^{2} \)
17 \( 1 + (-3.72 + 3.72i)T - 17iT^{2} \)
19 \( 1 - 4.09T + 19T^{2} \)
23 \( 1 + (2.12 - 2.12i)T - 23iT^{2} \)
29 \( 1 - 2.64T + 29T^{2} \)
31 \( 1 + 6.74T + 31T^{2} \)
37 \( 1 + (-0.822 - 0.822i)T + 37iT^{2} \)
41 \( 1 + 3.55iT - 41T^{2} \)
43 \( 1 + (5.07 + 5.07i)T + 43iT^{2} \)
47 \( 1 + (2.60 + 2.60i)T + 47iT^{2} \)
53 \( 1 + (-1.04 + 1.04i)T - 53iT^{2} \)
59 \( 1 - 1.60iT - 59T^{2} \)
61 \( 1 - 6.93iT - 61T^{2} \)
67 \( 1 + (-1.31 - 1.31i)T + 67iT^{2} \)
71 \( 1 - 2.87T + 71T^{2} \)
73 \( 1 + (10.4 + 10.4i)T + 73iT^{2} \)
79 \( 1 + 15.2T + 79T^{2} \)
83 \( 1 + (4.72 + 4.72i)T + 83iT^{2} \)
89 \( 1 + 11.1iT - 89T^{2} \)
97 \( 1 + (-7.11 - 7.11i)T + 97iT^{2} \)
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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

−11.63653624570833416259749479813, −10.32099440674554487077928149285, −9.494255952192826904209144977166, −8.638816208907074685988766297054, −7.66271484268326650947453688472, −7.20766432557723829292000053333, −5.72188912854487112846628954824, −4.91205573171419004234681693495, −3.64803699465242707603584633205, −2.82613033829196064193892570980, 0.02341734370237194262147948903, 1.28715659719239707547506997884, 3.17063426539062058755591889348, 4.22035224239392818466044263423, 5.54977070434708273663422962901, 6.55111471275362773128435685600, 7.10777394281356987388509849939, 8.173106550651117773165212475436, 9.441403355460285489956526370460, 10.01297987643286137676331805259

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