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Properties

Label 2-605-55.13-c1-0-39
Degree $2$
Conductor $605$
Sign $-0.919 - 0.392i$
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  + (−1.30 + 0.665i)2-s + (−0.822 − 0.130i)3-s + (0.0875 − 0.120i)4-s + (−0.233 − 2.22i)5-s + (1.16 − 0.377i)6-s + (−0.659 − 4.16i)7-s + (0.424 − 2.68i)8-s + (−2.19 − 0.712i)9-s + (1.78 + 2.74i)10-s + (−0.0877 + 0.0877i)12-s + (0.420 + 0.824i)13-s + (3.63 + 4.99i)14-s + (−0.0979 + 1.85i)15-s + (1.32 + 4.06i)16-s + (−0.875 + 1.71i)17-s + (3.34 − 0.529i)18-s + ⋯
L(s)  = 1  + (−0.923 + 0.470i)2-s + (−0.474 − 0.0751i)3-s + (0.0437 − 0.0602i)4-s + (−0.104 − 0.994i)5-s + (0.473 − 0.153i)6-s + (−0.249 − 1.57i)7-s + (0.150 − 0.947i)8-s + (−0.731 − 0.237i)9-s + (0.564 + 0.869i)10-s + (−0.0253 + 0.0253i)12-s + (0.116 + 0.228i)13-s + (0.970 + 1.33i)14-s + (−0.0252 + 0.479i)15-s + (0.330 + 1.01i)16-s + (−0.212 + 0.416i)17-s + (0.787 − 0.124i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.919 - 0.392i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.919 - 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.919 - 0.392i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.919 - 0.392i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00700310 + 0.0342484i\)
\(L(\frac12)\) \(\approx\) \(0.00700310 + 0.0342484i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.233 + 2.22i)T \)
11 \( 1 \)
good2 \( 1 + (1.30 - 0.665i)T + (1.17 - 1.61i)T^{2} \)
3 \( 1 + (0.822 + 0.130i)T + (2.85 + 0.927i)T^{2} \)
7 \( 1 + (0.659 + 4.16i)T + (-6.65 + 2.16i)T^{2} \)
13 \( 1 + (-0.420 - 0.824i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (0.875 - 1.71i)T + (-9.99 - 13.7i)T^{2} \)
19 \( 1 + (4.39 - 3.19i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-1.95 - 1.95i)T + 23iT^{2} \)
29 \( 1 + (-0.810 - 0.588i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.131 + 0.403i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (4.87 - 0.771i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (0.339 + 0.467i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + (-5.05 + 5.05i)T - 43iT^{2} \)
47 \( 1 + (0.186 - 1.17i)T + (-44.6 - 14.5i)T^{2} \)
53 \( 1 + (-8.09 + 4.12i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (5.47 - 7.53i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (7.40 - 2.40i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + (3.05 - 3.05i)T - 67iT^{2} \)
71 \( 1 + (-2.65 - 8.17i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (5.39 - 0.854i)T + (69.4 - 22.5i)T^{2} \)
79 \( 1 + (0.705 - 2.17i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.77 + 0.902i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + 13.9iT - 89T^{2} \)
97 \( 1 + (-1.50 - 2.96i)T + (-57.0 + 78.4i)T^{2} \)
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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

−10.14382451950367151178602985162, −9.043680363590678571086242293644, −8.498026914755046185423726464231, −7.57756376812068018483438928449, −6.77877808360679536456749025542, −5.80723874519734828444067508103, −4.41906627034415423408468530744, −3.68708179576497800988258065918, −1.21196350895557407146672307483, −0.03102957122990552734210532822, 2.27255252967304630796326189170, 2.90824596945471983471592041389, 4.87588684597371856060416235786, 5.80069082954911437016539237825, 6.55047854403354174551373879900, 7.924032110495926937079591734087, 8.794036988969581409908061383468, 9.318720680977620113706905467743, 10.47541999872515543403532021887, 10.95220557214777442068399562593

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