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Properties

Label 2-605-55.49-c1-0-12
Degree $2$
Conductor $605$
Sign $-0.0335 - 0.999i$
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  + (−0.312 + 0.101i)2-s + (0.565 − 0.778i)3-s + (−1.53 + 1.11i)4-s + (1.86 + 1.22i)5-s + (−0.0976 + 0.300i)6-s + (1.92 + 2.64i)7-s + (0.751 − 1.03i)8-s + (0.641 + 1.97i)9-s + (−0.708 − 0.194i)10-s + 1.82i·12-s + (−4.95 + 1.60i)13-s + (−0.869 − 0.631i)14-s + (2.01 − 0.759i)15-s + (1.03 − 3.20i)16-s + (−3.42 − 1.11i)17-s + (−0.400 − 0.551i)18-s + ⋯
L(s)  = 1  + (−0.220 + 0.0717i)2-s + (0.326 − 0.449i)3-s + (−0.765 + 0.556i)4-s + (0.835 + 0.549i)5-s + (−0.0398 + 0.122i)6-s + (0.727 + 1.00i)7-s + (0.265 − 0.365i)8-s + (0.213 + 0.657i)9-s + (−0.223 − 0.0613i)10-s + 0.525i·12-s + (−1.37 + 0.446i)13-s + (−0.232 − 0.168i)14-s + (0.519 − 0.196i)15-s + (0.259 − 0.800i)16-s + (−0.830 − 0.269i)17-s + (−0.0943 − 0.129i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0335 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0335 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.0335 - 0.999i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.0335 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.909434 + 0.940453i\)
\(L(\frac12)\) \(\approx\) \(0.909434 + 0.940453i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.86 - 1.22i)T \)
11 \( 1 \)
good2 \( 1 + (0.312 - 0.101i)T + (1.61 - 1.17i)T^{2} \)
3 \( 1 + (-0.565 + 0.778i)T + (-0.927 - 2.85i)T^{2} \)
7 \( 1 + (-1.92 - 2.64i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (4.95 - 1.60i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (3.42 + 1.11i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (4.05 + 2.94i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 5.84iT - 23T^{2} \)
29 \( 1 + (-1.65 + 1.20i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.00 + 3.09i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-3.66 - 5.04i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-8.25 - 5.99i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 0.596iT - 43T^{2} \)
47 \( 1 + (-0.565 + 0.778i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.374 - 0.121i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (0.426 - 0.310i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-1.64 + 5.06i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 8.45iT - 67T^{2} \)
71 \( 1 + (-3.87 + 11.9i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-5.00 - 6.88i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-0.875 - 2.69i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-11.5 - 3.75i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 + (-10.7 + 3.49i)T + (78.4 - 57.0i)T^{2} \)
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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

−10.87454716949683569919295584270, −9.637793331078016085717083848518, −9.222137356557030123587461581649, −8.201284638919715318383146731554, −7.51095849253837737025890854025, −6.56148051662699153875697920899, −5.20869578922445536302876225303, −4.54226768557392707617234857385, −2.71547126916435136574800872375, −2.01807230061716045843740068274, 0.77409448864445526498216297278, 2.21679310561848818459487032022, 4.19683338488828103074870681381, 4.58660026557143049356740401221, 5.68477337284973723105860928252, 6.82746896197595383914927274458, 8.100192478038773102169672972436, 8.872061868519079410913784122289, 9.566429331730027301999566558536, 10.41018200384380419427192335344

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