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Properties

Label 2-605-55.4-c1-0-18
Degree $2$
Conductor $605$
Sign $-0.790 - 0.612i$
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  + (−1.18 + 1.63i)2-s + (2.77 + 0.900i)3-s + (−0.646 − 1.98i)4-s + (0.135 + 2.23i)5-s + (−4.76 + 3.46i)6-s + (3.05 − 0.992i)7-s + (0.176 + 0.0573i)8-s + (4.44 + 3.23i)9-s + (−3.81 − 2.43i)10-s − 6.09i·12-s + (−0.381 + 0.524i)13-s + (−2.00 + 6.17i)14-s + (−1.63 + 6.30i)15-s + (3.08 − 2.23i)16-s + (0.698 + 0.961i)17-s + (−10.5 + 3.43i)18-s + ⋯
L(s)  = 1  + (−0.840 + 1.15i)2-s + (1.60 + 0.520i)3-s + (−0.323 − 0.994i)4-s + (0.0607 + 0.998i)5-s + (−1.94 + 1.41i)6-s + (1.15 − 0.375i)7-s + (0.0624 + 0.0202i)8-s + (1.48 + 1.07i)9-s + (−1.20 − 0.768i)10-s − 1.76i·12-s + (−0.105 + 0.145i)13-s + (−0.536 + 1.65i)14-s + (−0.421 + 1.62i)15-s + (0.770 − 0.559i)16-s + (0.169 + 0.233i)17-s + (−2.49 + 0.809i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.790 - 0.612i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.790 - 0.612i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (444, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.790 - 0.612i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.574748 + 1.68019i\)
\(L(\frac12)\) \(\approx\) \(0.574748 + 1.68019i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.135 - 2.23i)T \)
11 \( 1 \)
good2 \( 1 + (1.18 - 1.63i)T + (-0.618 - 1.90i)T^{2} \)
3 \( 1 + (-2.77 - 0.900i)T + (2.42 + 1.76i)T^{2} \)
7 \( 1 + (-3.05 + 0.992i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (0.381 - 0.524i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.698 - 0.961i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.584 - 1.79i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 4.35iT - 23T^{2} \)
29 \( 1 + (-1.28 - 3.96i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (6.39 + 4.64i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.95 + 0.636i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.02 - 3.14i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 10.9iT - 43T^{2} \)
47 \( 1 + (-2.77 - 0.900i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-3.77 + 5.19i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.14 - 3.53i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-6.29 + 4.57i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 2.37iT - 67T^{2} \)
71 \( 1 + (12.7 - 9.23i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (5.43 - 1.76i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (12.3 + 8.96i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (6.25 + 8.60i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 + (-8.89 + 12.2i)T + (-29.9 - 92.2i)T^{2} \)
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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

−10.42814097167475685630625151049, −9.963043542342005373301358549155, −8.854893321699758970757232683068, −8.406968646794718713033651185100, −7.52361340235985442567814417389, −7.12464381247814834263143448535, −5.78411416276428091195481018920, −4.35455125571913382750100064384, −3.29479532207690743240600380791, −2.02221323506045293003543024860, 1.25611839201433283776978607852, 2.00141577914087174975622285702, 3.01345924102800128399008394941, 4.25890037597681258055146124403, 5.56517469762431572969445314539, 7.40495005930287711357497968919, 8.135855959710424777089540233593, 8.705018064870302786352253875430, 9.255736171014432145935027543313, 10.00524131814761676045341302541

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