Properties

Label 2-605-55.18-c1-0-40
Degree $2$
Conductor $605$
Sign $-0.692 + 0.721i$
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  + (−0.229 − 1.44i)2-s + (0.741 − 0.377i)3-s + (−0.141 + 0.0460i)4-s + (2.04 − 0.908i)5-s + (−0.717 − 0.987i)6-s + (1.91 − 3.75i)7-s + (−1.23 − 2.41i)8-s + (−1.35 + 1.86i)9-s + (−1.78 − 2.74i)10-s + (−0.0877 + 0.0877i)12-s + (−0.914 + 0.144i)13-s + (−5.87 − 1.90i)14-s + (1.17 − 1.44i)15-s + (−3.45 + 2.51i)16-s + (1.90 + 0.301i)17-s + (3.01 + 1.53i)18-s + ⋯
L(s)  = 1  + (−0.162 − 1.02i)2-s + (0.428 − 0.218i)3-s + (−0.0708 + 0.0230i)4-s + (0.913 − 0.406i)5-s + (−0.292 − 0.403i)6-s + (0.723 − 1.41i)7-s + (−0.435 − 0.854i)8-s + (−0.452 + 0.622i)9-s + (−0.564 − 0.869i)10-s + (−0.0253 + 0.0253i)12-s + (−0.253 + 0.0401i)13-s + (−1.57 − 0.510i)14-s + (0.302 − 0.373i)15-s + (−0.864 + 0.628i)16-s + (0.461 + 0.0731i)17-s + (0.710 + 0.361i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.692 + 0.721i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.692 + 0.721i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.692 + 0.721i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.778242 - 1.82620i\)
\(L(\frac12)\) \(\approx\) \(0.778242 - 1.82620i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.04 + 0.908i)T \)
11 \( 1 \)
good2 \( 1 + (0.229 + 1.44i)T + (-1.90 + 0.618i)T^{2} \)
3 \( 1 + (-0.741 + 0.377i)T + (1.76 - 2.42i)T^{2} \)
7 \( 1 + (-1.91 + 3.75i)T + (-4.11 - 5.66i)T^{2} \)
13 \( 1 + (0.914 - 0.144i)T + (12.3 - 4.01i)T^{2} \)
17 \( 1 + (-1.90 - 0.301i)T + (16.1 + 5.25i)T^{2} \)
19 \( 1 + (1.67 - 5.16i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-1.95 - 1.95i)T + 23iT^{2} \)
29 \( 1 + (-0.309 - 0.952i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.342 + 0.249i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-4.39 - 2.23i)T + (21.7 + 29.9i)T^{2} \)
41 \( 1 + (0.549 + 0.178i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (5.05 - 5.05i)T - 43iT^{2} \)
47 \( 1 + (0.540 + 1.05i)T + (-27.6 + 38.0i)T^{2} \)
53 \( 1 + (1.42 + 8.96i)T + (-50.4 + 16.3i)T^{2} \)
59 \( 1 + (-8.85 + 2.87i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-4.57 - 6.29i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + (3.05 - 3.05i)T - 67iT^{2} \)
71 \( 1 + (6.95 - 5.04i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (4.86 + 2.47i)T + (42.9 + 59.0i)T^{2} \)
79 \( 1 + (1.84 + 1.34i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (0.311 - 1.96i)T + (-78.9 - 25.6i)T^{2} \)
89 \( 1 + 13.9iT - 89T^{2} \)
97 \( 1 + (-3.28 + 0.520i)T + (92.2 - 29.9i)T^{2} \)
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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

−10.24670967245106454887892948658, −9.891041576004348363177696790229, −8.671441019146513792617731064897, −7.82986337437973555522518577136, −6.86223205929417394775467413054, −5.65231172100833931489214160654, −4.49574076965304038429474300720, −3.29851386616257362628838505397, −2.03336919185830684714815717767, −1.21795629890521742410407175536, 2.25535114066218764977986491426, 2.92539330004363233984366202506, 4.92066090713115847787104521173, 5.73114002831706004536770511135, 6.39091122170523758314580420900, 7.38274442855381364963811974929, 8.563482560456250558607400079445, 8.883929041834832175323732874970, 9.746589053125137767889531452336, 11.04140061034233539169504441148

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