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Properties

Label 2-605-55.32-c1-0-8
Degree $2$
Conductor $605$
Sign $-0.655 - 0.755i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.655 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.655 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.655 - 0.755i$
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: \((2,\ 605,\ (\ :1/2),\ -0.655 - 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.364032 + 0.797404i\)
\(L(\frac12)\) \(\approx\) \(0.364032 + 0.797404i\)
\(L(\frac{3}{2})\) 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

−11.16247786154390228439797232509, −10.59649942366885964335123782065, −9.064511470387725263174932623561, −8.135044764775652018507409941513, −7.70361167023791057737059564687, −6.57407986342850975244092189770, −5.04031093887339567003333847054, −4.19123476431040735639791759086, −3.82108409522325554337362881181, −1.96113431573570686827673205155, 0.47036276895506745524001796732, 2.10684142602976516850681637972, 3.83705277468178342951083421130, 4.91283063382061315906858942745, 5.82037488788201848144557793730, 6.63196658234919471653227618797, 7.47213751918136902259537620822, 8.579245430091363723070319941616, 9.316211775566107937910359139178, 10.79590774757326125355467316973

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