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Properties

Label 2-605-11.4-c1-0-8
Degree $2$
Conductor $605$
Sign $-0.927 - 0.374i$
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  + (1.51 + 1.10i)2-s + (−0.549 + 1.69i)3-s + (0.469 + 1.44i)4-s + (−0.809 + 0.587i)5-s + (−2.69 + 1.96i)6-s + (1.31 + 4.04i)7-s + (0.278 − 0.857i)8-s + (−0.128 − 0.0937i)9-s − 1.87·10-s − 2.70·12-s + (−1.10 − 0.799i)13-s + (−2.46 + 7.59i)14-s + (−0.549 − 1.69i)15-s + (3.82 − 2.78i)16-s + (−1.69 + 1.23i)17-s + (−0.0924 − 0.284i)18-s + ⋯
L(s)  = 1  + (1.07 + 0.779i)2-s + (−0.317 + 0.976i)3-s + (0.234 + 0.722i)4-s + (−0.361 + 0.262i)5-s + (−1.10 + 0.800i)6-s + (0.497 + 1.53i)7-s + (0.0984 − 0.303i)8-s + (−0.0429 − 0.0312i)9-s − 0.593·10-s − 0.779·12-s + (−0.305 − 0.221i)13-s + (−0.659 + 2.03i)14-s + (−0.141 − 0.436i)15-s + (0.956 − 0.695i)16-s + (−0.411 + 0.299i)17-s + (−0.0217 − 0.0670i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 - 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.927 - 0.374i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.927 - 0.374i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.418796 + 2.15613i\)
\(L(\frac12)\) \(\approx\) \(0.418796 + 2.15613i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 \)
good2 \( 1 + (-1.51 - 1.10i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.549 - 1.69i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + (-1.31 - 4.04i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (1.10 + 0.799i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.69 - 1.23i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.186 + 0.574i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 4.39T + 23T^{2} \)
29 \( 1 + (2.04 + 6.30i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.77 - 1.29i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.90 - 5.86i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.28 + 7.04i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 12.6T + 43T^{2} \)
47 \( 1 + (0.950 - 2.92i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-5.38 - 3.91i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.71 - 11.4i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-4.60 + 3.34i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 9.86T + 67T^{2} \)
71 \( 1 + (-4.23 + 3.07i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.223 + 0.687i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (4.58 + 3.33i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-0.770 + 0.559i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 1.24T + 89T^{2} \)
97 \( 1 + (9.33 + 6.78i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.15713153724340663472627950704, −10.17738346582864632015016846753, −9.351259633076144105797957514153, −8.263566683221980509542253424111, −7.32966601781696509794767912154, −6.05161132073247936462434930496, −5.54495939708067059920138203274, −4.62121786445294360211852377250, −3.93265379115151512710304969610, −2.47997483266997600599060637031, 0.961068608546214148341095418066, 2.16515163151313818508549498812, 3.79430728235839605845852273783, 4.34480881724596366347082329615, 5.43337395806317240063452881615, 6.69765645159551535972999937252, 7.48742770214169434321128403985, 8.175925896270345549869316173108, 9.685810914864413952241761497540, 10.77044751685889236338123815825

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