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Properties

Label 2-605-55.43-c1-0-32
Degree $2$
Conductor $605$
Sign $-0.301 - 0.953i$
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  + (−1.63 − 1.63i)2-s + (−0.722 − 0.722i)3-s + 3.35i·4-s + (0.360 − 2.20i)5-s + 2.36i·6-s + (2.14 + 2.14i)7-s + (2.22 − 2.22i)8-s − 1.95i·9-s + (−4.20 + 3.02i)10-s + (2.42 − 2.42i)12-s + (−2.96 + 2.96i)13-s − 7.02i·14-s + (−1.85 + 1.33i)15-s − 0.554·16-s + (−4.75 − 4.75i)17-s + (−3.20 + 3.20i)18-s + ⋯
L(s)  = 1  + (−1.15 − 1.15i)2-s + (−0.417 − 0.417i)3-s + 1.67i·4-s + (0.161 − 0.986i)5-s + 0.965i·6-s + (0.811 + 0.811i)7-s + (0.785 − 0.785i)8-s − 0.651i·9-s + (−1.32 + 0.955i)10-s + (0.700 − 0.700i)12-s + (−0.821 + 0.821i)13-s − 1.87i·14-s + (−0.479 + 0.344i)15-s − 0.138·16-s + (−1.15 − 1.15i)17-s + (−0.754 + 0.754i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.301 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.301 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.301 - 0.953i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (483, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.301 - 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.153849 + 0.209895i\)
\(L(\frac12)\) \(\approx\) \(0.153849 + 0.209895i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.360 + 2.20i)T \)
11 \( 1 \)
good2 \( 1 + (1.63 + 1.63i)T + 2iT^{2} \)
3 \( 1 + (0.722 + 0.722i)T + 3iT^{2} \)
7 \( 1 + (-2.14 - 2.14i)T + 7iT^{2} \)
13 \( 1 + (2.96 - 2.96i)T - 13iT^{2} \)
17 \( 1 + (4.75 + 4.75i)T + 17iT^{2} \)
19 \( 1 - 0.733T + 19T^{2} \)
23 \( 1 + (4.96 + 4.96i)T + 23iT^{2} \)
29 \( 1 + 4.84T + 29T^{2} \)
31 \( 1 - 1.08T + 31T^{2} \)
37 \( 1 + (-0.0598 + 0.0598i)T - 37iT^{2} \)
41 \( 1 - 5.36iT - 41T^{2} \)
43 \( 1 + (-1.92 + 1.92i)T - 43iT^{2} \)
47 \( 1 + (7.22 - 7.22i)T - 47iT^{2} \)
53 \( 1 + (-1.60 - 1.60i)T + 53iT^{2} \)
59 \( 1 - 1.19iT - 59T^{2} \)
61 \( 1 + 7.37iT - 61T^{2} \)
67 \( 1 + (-3.89 + 3.89i)T - 67iT^{2} \)
71 \( 1 - 13.9T + 71T^{2} \)
73 \( 1 + (1.92 - 1.92i)T - 73iT^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 + (1.21 - 1.21i)T - 83iT^{2} \)
89 \( 1 - 10.9iT - 89T^{2} \)
97 \( 1 + (3.15 - 3.15i)T - 97iT^{2} \)
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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

−9.754788794557965692497164158207, −9.313946985510247369023228994260, −8.617829344633858107944390691649, −7.79226204994779262665150540531, −6.56418326310199213680945455437, −5.30507311269602531880641121140, −4.27806514750708758573090256695, −2.48601595942807273232385384156, −1.64112167722323944342067713851, −0.21629364079977227148878091174, 1.95408349844690061304466077084, 3.91854655676399812673665680370, 5.22830507094066122358425771331, 6.02288356560841695662292865126, 7.09912685707575076069658713365, 7.69000732052999948670987924064, 8.331551272309855088981016344479, 9.657183828223594606912978981772, 10.30712927859930703828785948450, 10.78050116303231489240212515997

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