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Properties

Label 2-605-55.14-c1-0-40
Degree $2$
Conductor $605$
Sign $0.909 - 0.415i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.909 - 0.415i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.909 - 0.415i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.41548 + 0.742897i\)
\(L(\frac12)\) \(\approx\) \(3.41548 + 0.742897i\)
\(L(\frac{3}{2})\) 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

−10.40899076849599406362546258367, −9.345379603242672827637187586376, −8.828756683912244011479409031668, −7.81351705926394799690944600300, −7.24111469342487040633245797626, −6.33442387832851932793316185177, −5.26856585398520617400959693203, −3.97168594911480681143678985880, −3.37355742240321309732916908105, −1.63224929358317049565308522738, 2.30087185034234083307882526422, 2.80116725899669152187751991832, 3.57210583918603819743178438893, 4.34019916631711627294915652410, 5.86818055714126081399799873376, 7.10810140772418948883256360397, 8.069172558517929686472365688061, 9.311580347311903464472112553163, 9.752345597895808714623446627007, 10.57566804484184204589573102761

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