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Properties

Label 2-605-55.8-c1-0-35
Degree $2$
Conductor $605$
Sign $0.976 - 0.213i$
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  + (2.20 + 0.349i)2-s + (0.642 − 1.26i)3-s + (2.85 + 0.927i)4-s + (1.03 + 1.98i)5-s + (1.85 − 2.55i)6-s + (1.99 + 1.01i)8-s + (0.587 + 0.809i)9-s + (1.58 + 4.74i)10-s + (3.00 − 3.00i)12-s + (−0.699 + 4.41i)13-s + (3.16 − 0.0240i)15-s + (−0.809 − 0.587i)16-s + (−0.699 − 4.41i)17-s + (1.01 + 1.99i)18-s + (−1.95 − 6.01i)19-s + (1.09 + 6.61i)20-s + ⋯
L(s)  = 1  + (1.56 + 0.247i)2-s + (0.370 − 0.727i)3-s + (1.42 + 0.463i)4-s + (0.460 + 0.887i)5-s + (0.758 − 1.04i)6-s + (0.704 + 0.358i)8-s + (0.195 + 0.269i)9-s + (0.500 + 1.50i)10-s + (0.866 − 0.866i)12-s + (−0.194 + 1.22i)13-s + (0.816 − 0.00619i)15-s + (−0.202 − 0.146i)16-s + (−0.169 − 1.07i)17-s + (0.239 + 0.469i)18-s + (−0.448 − 1.37i)19-s + (0.245 + 1.47i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.213i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 - 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.976 - 0.213i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.976 - 0.213i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.93301 + 0.425710i\)
\(L(\frac12)\) \(\approx\) \(3.93301 + 0.425710i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.03 - 1.98i)T \)
11 \( 1 \)
good2 \( 1 + (-2.20 - 0.349i)T + (1.90 + 0.618i)T^{2} \)
3 \( 1 + (-0.642 + 1.26i)T + (-1.76 - 2.42i)T^{2} \)
7 \( 1 + (4.11 - 5.66i)T^{2} \)
13 \( 1 + (0.699 - 4.41i)T + (-12.3 - 4.01i)T^{2} \)
17 \( 1 + (0.699 + 4.41i)T + (-16.1 + 5.25i)T^{2} \)
19 \( 1 + (1.95 + 6.01i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (1 + i)T + 23iT^{2} \)
29 \( 1 + (-1.95 + 6.01i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.61 - 1.17i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.92 + 3.78i)T + (-21.7 + 29.9i)T^{2} \)
41 \( 1 + (6.01 - 1.95i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + (-3.78 - 1.92i)T + (27.6 + 38.0i)T^{2} \)
53 \( 1 + (-1.39 - 0.221i)T + (50.4 + 16.3i)T^{2} \)
59 \( 1 + (-5.70 - 1.85i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (3.71 - 5.11i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + (-3 + 3i)T - 67iT^{2} \)
71 \( 1 + (-6.47 - 4.70i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.03 - 3.98i)T + (-42.9 + 59.0i)T^{2} \)
79 \( 1 + (5.11 - 3.71i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (8.83 - 1.39i)T + (78.9 - 25.6i)T^{2} \)
89 \( 1 - 6iT - 89T^{2} \)
97 \( 1 + (-1.54 + 9.77i)T + (-92.2 - 29.9i)T^{2} \)
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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

−11.12534983024074315209541757857, −9.883269418944073710294297001696, −8.878436483645836552336682784218, −7.41334857758087255999364450764, −6.91189103136529334804640249623, −6.28851824391141586392786717415, −5.05822593838229836648650435858, −4.19509198913923852288417658663, −2.79404284393310502654472223110, −2.17724805185864803475161703889, 1.78605350298339992718596186574, 3.30073439022723888537533277950, 3.97389246238403615465089770037, 4.95951574540534895433040627384, 5.65702329152087153165038661716, 6.55898019551140390189596042124, 8.143981173046922473985095539587, 8.881861978690588735753783825414, 10.10354099226444221006589834938, 10.48647235861506752153048922853

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