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Properties

Label 2-605-55.2-c1-0-43
Degree $2$
Conductor $605$
Sign $0.100 - 0.994i$
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  + (−0.562 − 1.10i)2-s + (0.397 − 2.51i)3-s + (0.274 − 0.377i)4-s + (−0.841 + 2.07i)5-s + (−2.99 + 0.973i)6-s + (−3.42 + 0.542i)7-s + (−3.01 − 0.477i)8-s + (−3.30 − 1.07i)9-s + (2.75 − 0.235i)10-s + (−0.839 − 0.839i)12-s + (0.658 − 0.335i)13-s + (2.52 + 3.47i)14-s + (4.87 + 2.94i)15-s + (0.880 + 2.70i)16-s + (−0.550 − 0.280i)17-s + (0.672 + 4.24i)18-s + ⋯
L(s)  = 1  + (−0.397 − 0.780i)2-s + (0.229 − 1.45i)3-s + (0.137 − 0.188i)4-s + (−0.376 + 0.926i)5-s + (−1.22 + 0.397i)6-s + (−1.29 + 0.205i)7-s + (−1.06 − 0.168i)8-s + (−1.10 − 0.357i)9-s + (0.872 − 0.0744i)10-s + (−0.242 − 0.242i)12-s + (0.182 − 0.0931i)13-s + (0.674 + 0.928i)14-s + (1.25 + 0.759i)15-s + (0.220 + 0.677i)16-s + (−0.133 − 0.0679i)17-s + (0.158 + 1.00i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.100 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.100 - 0.994i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.100 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.151885 + 0.137286i\)
\(L(\frac12)\) \(\approx\) \(0.151885 + 0.137286i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.841 - 2.07i)T \)
11 \( 1 \)
good2 \( 1 + (0.562 + 1.10i)T + (-1.17 + 1.61i)T^{2} \)
3 \( 1 + (-0.397 + 2.51i)T + (-2.85 - 0.927i)T^{2} \)
7 \( 1 + (3.42 - 0.542i)T + (6.65 - 2.16i)T^{2} \)
13 \( 1 + (-0.658 + 0.335i)T + (7.64 - 10.5i)T^{2} \)
17 \( 1 + (0.550 + 0.280i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (2.54 - 1.84i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (4.30 - 4.30i)T - 23iT^{2} \)
29 \( 1 + (-2.34 - 1.70i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.937 + 2.88i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.359 + 2.27i)T + (-35.1 + 11.4i)T^{2} \)
41 \( 1 + (0.599 + 0.825i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + (4.07 + 4.07i)T + 43iT^{2} \)
47 \( 1 + (1.07 + 0.169i)T + (44.6 + 14.5i)T^{2} \)
53 \( 1 + (1.94 + 3.82i)T + (-31.1 + 42.8i)T^{2} \)
59 \( 1 + (4.35 - 5.98i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (3.56 - 1.15i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + (9.39 + 9.39i)T + 67iT^{2} \)
71 \( 1 + (-1.11 - 3.43i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.0841 + 0.531i)T + (-69.4 + 22.5i)T^{2} \)
79 \( 1 + (-3.77 + 11.6i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-7.27 + 14.2i)T + (-48.7 - 67.1i)T^{2} \)
89 \( 1 - 14.6iT - 89T^{2} \)
97 \( 1 + (12.7 - 6.51i)T + (57.0 - 78.4i)T^{2} \)
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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

−10.11921392925993031667606005182, −9.256093811327245397952144158126, −8.163734405210349715506747131378, −7.22749563245331860185229551668, −6.41145937432916918401532682003, −5.99002970930395836634864944523, −3.60169398231956495058657285465, −2.76192006716081017793036993738, −1.80843566661477191970251615198, −0.11605414078106969237493192097, 2.93757346700546618216514145624, 3.88604119745475638966319272100, 4.74719507464449948340221716204, 6.00664633408173330556170519990, 6.81607765022380186319598757155, 8.117861969640450474041438320336, 8.716080862274092585375129296354, 9.455867903392512197405192223408, 10.07734808896478396940661879967, 11.12107645428717573375195566619

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