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Properties

Label 2-605-55.4-c1-0-10
Degree $2$
Conductor $605$
Sign $0.364 - 0.931i$
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  + (0.725 − 0.998i)2-s + (−0.346 − 0.112i)3-s + (0.147 + 0.453i)4-s + (0.0238 + 2.23i)5-s + (−0.363 + 0.264i)6-s + (−2.45 + 0.798i)7-s + (2.90 + 0.944i)8-s + (−2.31 − 1.68i)9-s + (2.24 + 1.59i)10-s − 0.173i·12-s + (−1.62 + 2.23i)13-s + (−0.985 + 3.03i)14-s + (0.243 − 0.776i)15-s + (2.27 − 1.65i)16-s + (2.26 + 3.11i)17-s + (−3.36 + 1.09i)18-s + ⋯
L(s)  = 1  + (0.512 − 0.705i)2-s + (−0.199 − 0.0649i)3-s + (0.0737 + 0.226i)4-s + (0.0106 + 0.999i)5-s + (−0.148 + 0.107i)6-s + (−0.929 + 0.301i)7-s + (1.02 + 0.333i)8-s + (−0.773 − 0.561i)9-s + (0.711 + 0.505i)10-s − 0.0501i·12-s + (−0.449 + 0.619i)13-s + (−0.263 + 0.810i)14-s + (0.0628 − 0.200i)15-s + (0.569 − 0.414i)16-s + (0.549 + 0.755i)17-s + (−0.793 + 0.257i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.364 - 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.364 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.364 - 0.931i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (444, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.364 - 0.931i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15560 + 0.788697i\)
\(L(\frac12)\) \(\approx\) \(1.15560 + 0.788697i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.0238 - 2.23i)T \)
11 \( 1 \)
good2 \( 1 + (-0.725 + 0.998i)T + (-0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.346 + 0.112i)T + (2.42 + 1.76i)T^{2} \)
7 \( 1 + (2.45 - 0.798i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (1.62 - 2.23i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.26 - 3.11i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.0857 - 0.264i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 8.40iT - 23T^{2} \)
29 \( 1 + (-1.02 - 3.16i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.456 + 0.331i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.497 + 0.161i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.57 + 4.86i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 2.54iT - 43T^{2} \)
47 \( 1 + (4.68 + 1.52i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-5.12 + 7.05i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.31 + 7.13i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-11.4 + 8.33i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 3.20iT - 67T^{2} \)
71 \( 1 + (-6.79 + 4.93i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-12.3 + 4.02i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (7.85 + 5.70i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (1.93 + 2.66i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + 2.48T + 89T^{2} \)
97 \( 1 + (6.40 - 8.81i)T + (-29.9 - 92.2i)T^{2} \)
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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

−11.11566373528230640858286722855, −10.09254743409908518073584668135, −9.361538217443242632136572913921, −8.099376641647596180992787862385, −7.09907216060034536064213018859, −6.31656923139219805551835118140, −5.30876034423748204691118351264, −3.65518256414444666950983703560, −3.27506618589189055627159831232, −2.07652539280701263387904681000, 0.65524425543414335256715691785, 2.64904693899437268996673817454, 4.24518434585166654752993914487, 5.11577123454208759109913430012, 5.79263579267227378258600717715, 6.70221481271304323212819616866, 7.72851313353913350586078854851, 8.579828393167418704884050314835, 9.766636262107555356474197858877, 10.30278418653877759158274526792

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