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Properties

Label 2-605-11.5-c1-0-9
Degree $2$
Conductor $605$
Sign $-0.836 - 0.548i$
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  + (0.431 + 1.32i)2-s + (1.33 + 0.973i)3-s + (0.0415 − 0.0301i)4-s + (−0.309 + 0.951i)5-s + (−0.714 + 2.19i)6-s + (−3.80 + 2.76i)7-s + (2.31 + 1.68i)8-s + (−0.0801 − 0.246i)9-s − 1.39·10-s + 0.0850·12-s + (1.56 + 4.80i)13-s + (−5.31 − 3.86i)14-s + (−1.33 + 0.973i)15-s + (−1.20 + 3.70i)16-s + (1.64 − 5.05i)17-s + (0.293 − 0.212i)18-s + ⋯
L(s)  = 1  + (0.305 + 0.938i)2-s + (0.773 + 0.561i)3-s + (0.0207 − 0.0150i)4-s + (−0.138 + 0.425i)5-s + (−0.291 + 0.897i)6-s + (−1.43 + 1.04i)7-s + (0.819 + 0.595i)8-s + (−0.0267 − 0.0822i)9-s − 0.441·10-s + 0.0245·12-s + (0.432 + 1.33i)13-s + (−1.42 − 1.03i)14-s + (−0.345 + 0.251i)15-s + (−0.300 + 0.925i)16-s + (0.398 − 1.22i)17-s + (0.0690 − 0.0501i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 - 0.548i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.836 - 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.836 - 0.548i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.836 - 0.548i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.589117 + 1.97139i\)
\(L(\frac12)\) \(\approx\) \(0.589117 + 1.97139i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 \)
good2 \( 1 + (-0.431 - 1.32i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-1.33 - 0.973i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + (3.80 - 2.76i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-1.56 - 4.80i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.64 + 5.05i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.82 + 1.32i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 1.05T + 23T^{2} \)
29 \( 1 + (2.25 - 1.64i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.15 - 3.55i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.640 + 0.465i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-4.97 - 3.61i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 2.70T + 43T^{2} \)
47 \( 1 + (-9.09 - 6.60i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.324 - 0.999i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (3.66 - 2.66i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-3.05 + 9.39i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 10.4T + 67T^{2} \)
71 \( 1 + (0.324 - 0.999i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-8.17 + 5.93i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.90 + 8.95i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.974 - 2.99i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 12.5T + 89T^{2} \)
97 \( 1 + (5.49 + 16.9i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.97723559035218467363598261708, −9.742457172319817056848074552956, −9.251012298004908382260033168742, −8.477031199021133798382225742747, −7.17429584296106901071080228203, −6.52834562395912000035714720428, −5.80134391710673615973770774740, −4.52922603904136662753396727070, −3.33301622008485141316112701837, −2.43587914218885274630925865587, 0.975008654744383492572241252705, 2.42702680432739468572321571946, 3.47076318565961830118367718248, 4.00065259120375238615578181397, 5.73427833504594692441713571138, 6.90143436012931757414902819748, 7.71581892853032938162266017410, 8.423351791194752764617961281236, 9.710628143881564674637870696994, 10.42044526337001824181484550177

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