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Properties

Label 2-605-55.32-c1-0-2
Degree $2$
Conductor $605$
Sign $-0.931 - 0.364i$
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  + (0.895 − 0.895i)2-s + (−1.60 + 1.60i)3-s + 0.396i·4-s + (−0.452 − 2.18i)5-s + 2.87i·6-s + (−1.81 + 1.81i)7-s + (2.14 + 2.14i)8-s − 2.16i·9-s + (−2.36 − 1.55i)10-s + (−0.637 − 0.637i)12-s + (−3.52 − 3.52i)13-s + 3.24i·14-s + (4.24 + 2.79i)15-s + 3.04·16-s + (−0.212 + 0.212i)17-s + (−1.93 − 1.93i)18-s + ⋯
L(s)  = 1  + (0.633 − 0.633i)2-s + (−0.927 + 0.927i)3-s + 0.198i·4-s + (−0.202 − 0.979i)5-s + 1.17i·6-s + (−0.684 + 0.684i)7-s + (0.758 + 0.758i)8-s − 0.721i·9-s + (−0.747 − 0.492i)10-s + (−0.184 − 0.184i)12-s + (−0.977 − 0.977i)13-s + 0.867i·14-s + (1.09 + 0.721i)15-s + 0.762·16-s + (−0.0516 + 0.0516i)17-s + (−0.456 − 0.456i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.931 - 0.364i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.931 - 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.931 - 0.364i$
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: \((2,\ 605,\ (\ :1/2),\ -0.931 - 0.364i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0636349 + 0.337089i\)
\(L(\frac12)\) \(\approx\) \(0.0636349 + 0.337089i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.452 + 2.18i)T \)
11 \( 1 \)
good2 \( 1 + (-0.895 + 0.895i)T - 2iT^{2} \)
3 \( 1 + (1.60 - 1.60i)T - 3iT^{2} \)
7 \( 1 + (1.81 - 1.81i)T - 7iT^{2} \)
13 \( 1 + (3.52 + 3.52i)T + 13iT^{2} \)
17 \( 1 + (0.212 - 0.212i)T - 17iT^{2} \)
19 \( 1 + 6.08T + 19T^{2} \)
23 \( 1 + (5.07 - 5.07i)T - 23iT^{2} \)
29 \( 1 + 3.98T + 29T^{2} \)
31 \( 1 - 7.59T + 31T^{2} \)
37 \( 1 + (4.56 + 4.56i)T + 37iT^{2} \)
41 \( 1 + 2.42iT - 41T^{2} \)
43 \( 1 + (1.43 + 1.43i)T + 43iT^{2} \)
47 \( 1 + (-2.49 - 2.49i)T + 47iT^{2} \)
53 \( 1 + (6.21 - 6.21i)T - 53iT^{2} \)
59 \( 1 - 1.17iT - 59T^{2} \)
61 \( 1 - 0.0929iT - 61T^{2} \)
67 \( 1 + (-7.07 - 7.07i)T + 67iT^{2} \)
71 \( 1 - 11.2T + 71T^{2} \)
73 \( 1 + (-2.80 - 2.80i)T + 73iT^{2} \)
79 \( 1 + 2.35T + 79T^{2} \)
83 \( 1 + (5.86 + 5.86i)T + 83iT^{2} \)
89 \( 1 - 2.55iT - 89T^{2} \)
97 \( 1 + (-5.32 - 5.32i)T + 97iT^{2} \)
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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

−11.18770988837956625163386568972, −10.29771082381732961548343334524, −9.589801112623654608252141811696, −8.527150627902969960550160785059, −7.62885657544087818510026881628, −5.96993883923454344038281297986, −5.30214968584919214443408675270, −4.49900921913928422498457200050, −3.67370647903863372975844132866, −2.29674289033904330776024682419, 0.16126521349347227318033126977, 2.07522374802506629226884000019, 3.84188369052916367156589908580, 4.80897169124008370458452126295, 6.16836789278875671202897303986, 6.62303990521662689144693688380, 6.96484295929956216064973713194, 8.007487443319468295701594248748, 9.747602931069761458884640827371, 10.35037692944904464513697449742

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