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Properties

Label 2-605-55.32-c1-0-24
Degree $2$
Conductor $605$
Sign $0.856 - 0.516i$
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  + (−1.03 + 1.03i)2-s + (0.588 − 0.588i)3-s − 0.149i·4-s + (−2.18 + 0.465i)5-s + 1.22i·6-s + (2.98 − 2.98i)7-s + (−1.91 − 1.91i)8-s + 2.30i·9-s + (1.78 − 2.74i)10-s + (−0.0877 − 0.0877i)12-s + (−0.654 − 0.654i)13-s + 6.17i·14-s + (−1.01 + 1.56i)15-s + 4.27·16-s + (1.36 − 1.36i)17-s + (−2.39 − 2.39i)18-s + ⋯
L(s)  = 1  + (−0.732 + 0.732i)2-s + (0.339 − 0.339i)3-s − 0.0745i·4-s + (−0.978 + 0.208i)5-s + 0.498i·6-s + (1.12 − 1.12i)7-s + (−0.678 − 0.678i)8-s + 0.768i·9-s + (0.564 − 0.869i)10-s + (−0.0253 − 0.0253i)12-s + (−0.181 − 0.181i)13-s + 1.65i·14-s + (−0.261 + 0.403i)15-s + 1.06·16-s + (0.330 − 0.330i)17-s + (−0.563 − 0.563i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 - 0.516i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.856 - 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.856 - 0.516i$
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: \((2,\ 605,\ (\ :1/2),\ 0.856 - 0.516i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00671 + 0.279952i\)
\(L(\frac12)\) \(\approx\) \(1.00671 + 0.279952i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.18 - 0.465i)T \)
11 \( 1 \)
good2 \( 1 + (1.03 - 1.03i)T - 2iT^{2} \)
3 \( 1 + (-0.588 + 0.588i)T - 3iT^{2} \)
7 \( 1 + (-2.98 + 2.98i)T - 7iT^{2} \)
13 \( 1 + (0.654 + 0.654i)T + 13iT^{2} \)
17 \( 1 + (-1.36 + 1.36i)T - 17iT^{2} \)
19 \( 1 - 5.43T + 19T^{2} \)
23 \( 1 + (-1.95 + 1.95i)T - 23iT^{2} \)
29 \( 1 + 1.00T + 29T^{2} \)
31 \( 1 - 0.423T + 31T^{2} \)
37 \( 1 + (-3.48 - 3.48i)T + 37iT^{2} \)
41 \( 1 + 0.577iT - 41T^{2} \)
43 \( 1 + (-5.05 - 5.05i)T + 43iT^{2} \)
47 \( 1 + (-0.841 - 0.841i)T + 47iT^{2} \)
53 \( 1 + (-6.42 + 6.42i)T - 53iT^{2} \)
59 \( 1 - 9.30iT - 59T^{2} \)
61 \( 1 + 7.78iT - 61T^{2} \)
67 \( 1 + (3.05 + 3.05i)T + 67iT^{2} \)
71 \( 1 - 8.59T + 71T^{2} \)
73 \( 1 + (-3.86 - 3.86i)T + 73iT^{2} \)
79 \( 1 + 2.28T + 79T^{2} \)
83 \( 1 + (1.40 + 1.40i)T + 83iT^{2} \)
89 \( 1 - 13.9iT - 89T^{2} \)
97 \( 1 + (2.35 + 2.35i)T + 97iT^{2} \)
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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

−10.72905518854701269499222187074, −9.714903487016406944428677961353, −8.516772328796584450711739911319, −7.86967974610567147454201669903, −7.50616471457668559356841086257, −6.84889319021446498483452590348, −5.18640272892202415481173855737, −4.14901117846934901567676612557, −2.95534473560747022874494652327, −0.974370865422423617127782388557, 1.10481901649977372917543119052, 2.54024557276701708089658179413, 3.66674489479190041454590603538, 4.96464411557547827955961116450, 5.82471048436201115757931126201, 7.42917077803937747234870097019, 8.302666000480270405825065931800, 8.984138980040405301186475916478, 9.498703799339217169677734784468, 10.62831800690619422575305995519

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