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Properties

Label 2-605-55.9-c1-0-11
Degree $2$
Conductor $605$
Sign $0.815 - 0.578i$
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  + (−1.17 − 0.381i)2-s + (0.213 + 0.294i)3-s + (−0.386 − 0.280i)4-s + (2.13 + 0.668i)5-s + (−0.138 − 0.427i)6-s + (−1.51 + 2.09i)7-s + (1.79 + 2.47i)8-s + (0.886 − 2.72i)9-s + (−2.24 − 1.59i)10-s − 0.173i·12-s + (2.62 + 0.852i)13-s + (2.58 − 1.87i)14-s + (0.259 + 0.771i)15-s + (−0.870 − 2.67i)16-s + (−3.66 + 1.19i)17-s + (−2.07 + 2.86i)18-s + ⋯
L(s)  = 1  + (−0.829 − 0.269i)2-s + (0.123 + 0.170i)3-s + (−0.193 − 0.140i)4-s + (0.954 + 0.298i)5-s + (−0.0566 − 0.174i)6-s + (−0.574 + 0.790i)7-s + (0.635 + 0.874i)8-s + (0.295 − 0.909i)9-s + (−0.711 − 0.505i)10-s − 0.0501i·12-s + (0.727 + 0.236i)13-s + (0.689 − 0.501i)14-s + (0.0670 + 0.199i)15-s + (−0.217 − 0.669i)16-s + (−0.888 + 0.288i)17-s + (−0.490 + 0.674i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 - 0.578i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.815 - 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.815 - 0.578i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.815 - 0.578i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.943758 + 0.300749i\)
\(L(\frac12)\) \(\approx\) \(0.943758 + 0.300749i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.13 - 0.668i)T \)
11 \( 1 \)
good2 \( 1 + (1.17 + 0.381i)T + (1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.213 - 0.294i)T + (-0.927 + 2.85i)T^{2} \)
7 \( 1 + (1.51 - 2.09i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (-2.62 - 0.852i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (3.66 - 1.19i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.224 - 0.163i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 8.40iT - 23T^{2} \)
29 \( 1 + (-2.68 - 1.95i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.174 + 0.536i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.307 - 0.422i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-4.13 + 3.00i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 2.54iT - 43T^{2} \)
47 \( 1 + (-2.89 - 3.98i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-8.29 - 2.69i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-6.07 - 4.41i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-4.38 - 13.4i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 3.20iT - 67T^{2} \)
71 \( 1 + (2.59 + 7.98i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-7.66 + 10.5i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.99 - 9.22i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-3.13 + 1.01i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + 2.48T + 89T^{2} \)
97 \( 1 + (10.3 + 3.36i)T + (78.4 + 57.0i)T^{2} \)
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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

−10.47009245582714653968738899935, −9.696936826838191801867602259554, −9.102039164424544665537152097831, −8.676930589637603829218086859875, −7.18940975537777898468882414741, −6.15782261186553332622144232525, −5.47293092184164476802827324213, −3.99704695783039235576533051414, −2.62853898682295476354411822717, −1.37247929252557407110064193124, 0.821801382694728616425038685287, 2.37855466272512933961432159777, 4.02112898670369363404910380281, 4.96354388819002155391990038677, 6.42986642944743748212461245519, 7.00192892139051912244065810365, 8.180604685667047308047615168824, 8.703654334867829488550454090944, 9.712449413143777927838190954708, 10.31417273513382704924108174528

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