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Properties

Label 2-605-55.3-c0-0-0
Degree $2$
Conductor $605$
Sign $-0.442 - 0.896i$
Analytic cond. $0.301934$
Root an. cond. $0.549485$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.642 + 1.26i)3-s + (0.951 + 0.309i)4-s + (−0.587 + 0.809i)5-s + (−0.587 − 0.809i)9-s + (−1 + i)12-s + (−0.642 − 1.26i)15-s + (0.809 + 0.587i)16-s + (−0.809 + 0.587i)20-s + (−1 − i)23-s + (−0.309 − 0.951i)25-s + (−0.309 − 0.951i)36-s + (0.642 + 1.26i)37-s + 1.00·45-s + (1.26 + 0.642i)47-s + (−1.26 + 0.642i)48-s + (0.587 − 0.809i)49-s + ⋯
L(s)  = 1  + (−0.642 + 1.26i)3-s + (0.951 + 0.309i)4-s + (−0.587 + 0.809i)5-s + (−0.587 − 0.809i)9-s + (−1 + i)12-s + (−0.642 − 1.26i)15-s + (0.809 + 0.587i)16-s + (−0.809 + 0.587i)20-s + (−1 − i)23-s + (−0.309 − 0.951i)25-s + (−0.309 − 0.951i)36-s + (0.642 + 1.26i)37-s + 1.00·45-s + (1.26 + 0.642i)47-s + (−1.26 + 0.642i)48-s + (0.587 − 0.809i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.442 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.442 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.442 - 0.896i$
Analytic conductor: \(0.301934\)
Root analytic conductor: \(0.549485\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :0),\ -0.442 - 0.896i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8159426242\)
\(L(\frac12)\) \(\approx\) \(0.8159426242\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.587 - 0.809i)T \)
11 \( 1 \)
good2 \( 1 + (-0.951 - 0.309i)T^{2} \)
3 \( 1 + (0.642 - 1.26i)T + (-0.587 - 0.809i)T^{2} \)
7 \( 1 + (-0.587 + 0.809i)T^{2} \)
13 \( 1 + (0.951 + 0.309i)T^{2} \)
17 \( 1 + (0.951 - 0.309i)T^{2} \)
19 \( 1 + (0.809 - 0.587i)T^{2} \)
23 \( 1 + (1 + i)T + iT^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (0.309 - 0.951i)T^{2} \)
37 \( 1 + (-0.642 - 1.26i)T + (-0.587 + 0.809i)T^{2} \)
41 \( 1 + (-0.809 + 0.587i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-1.26 - 0.642i)T + (0.587 + 0.809i)T^{2} \)
53 \( 1 + (-1.39 - 0.221i)T + (0.951 + 0.309i)T^{2} \)
59 \( 1 + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.309 + 0.951i)T^{2} \)
67 \( 1 + (-1 + i)T - iT^{2} \)
71 \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.587 - 0.809i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.951 + 0.309i)T^{2} \)
89 \( 1 - 2iT - T^{2} \)
97 \( 1 + (-0.221 + 1.39i)T + (-0.951 - 0.309i)T^{2} \)
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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

−10.96416955774581564923394949824, −10.50681323268341012871413717064, −9.806823773817225625560878366607, −8.459186562246442588866754211121, −7.53666071087529629579232398598, −6.56876056037991456570817729949, −5.78501667323199032752383553393, −4.45377581246930556150005202452, −3.64250115166442234441480454658, −2.51612685095028633135248256053, 1.08259507175776341662098159118, 2.24008365969667372946248075787, 3.94135349979347231119537022174, 5.48791725371265767989687517071, 6.00101410166442674505115318748, 7.22926784594741112986842755403, 7.52277158992579536781293853955, 8.606061921058946658136938767838, 9.811973448822862727553915900723, 10.92087499033879831073394419719

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