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Properties

Label 2-605-55.32-c1-0-36
Degree $2$
Conductor $605$
Sign $-0.381 + 0.924i$
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  + (−1.76 + 1.76i)2-s + (1.85 − 1.85i)3-s − 4.25i·4-s + (−2.19 + 0.412i)5-s + 6.56i·6-s + (0.260 − 0.260i)7-s + (3.98 + 3.98i)8-s − 3.88i·9-s + (3.15 − 4.61i)10-s + (−7.89 − 7.89i)12-s + (−3.13 − 3.13i)13-s + 0.922i·14-s + (−3.31 + 4.84i)15-s − 5.59·16-s + (0.608 − 0.608i)17-s + (6.87 + 6.87i)18-s + ⋯
L(s)  = 1  + (−1.25 + 1.25i)2-s + (1.07 − 1.07i)3-s − 2.12i·4-s + (−0.982 + 0.184i)5-s + 2.67i·6-s + (0.0985 − 0.0985i)7-s + (1.41 + 1.41i)8-s − 1.29i·9-s + (0.998 − 1.45i)10-s + (−2.27 − 2.27i)12-s + (−0.868 − 0.868i)13-s + 0.246i·14-s + (−0.855 + 1.25i)15-s − 1.39·16-s + (0.147 − 0.147i)17-s + (1.62 + 1.62i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.381 + 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.381 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.381 + 0.924i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (362, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.381 + 0.924i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.215752 - 0.322401i\)
\(L(\frac12)\) \(\approx\) \(0.215752 - 0.322401i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.19 - 0.412i)T \)
11 \( 1 \)
good2 \( 1 + (1.76 - 1.76i)T - 2iT^{2} \)
3 \( 1 + (-1.85 + 1.85i)T - 3iT^{2} \)
7 \( 1 + (-0.260 + 0.260i)T - 7iT^{2} \)
13 \( 1 + (3.13 + 3.13i)T + 13iT^{2} \)
17 \( 1 + (-0.608 + 0.608i)T - 17iT^{2} \)
19 \( 1 + 4.44T + 19T^{2} \)
23 \( 1 + (2.17 - 2.17i)T - 23iT^{2} \)
29 \( 1 + 0.769T + 29T^{2} \)
31 \( 1 + 4.53T + 31T^{2} \)
37 \( 1 + (7.75 + 7.75i)T + 37iT^{2} \)
41 \( 1 - 5.01iT - 41T^{2} \)
43 \( 1 + (4.69 + 4.69i)T + 43iT^{2} \)
47 \( 1 + (7.64 + 7.64i)T + 47iT^{2} \)
53 \( 1 + (-0.711 + 0.711i)T - 53iT^{2} \)
59 \( 1 + 1.09iT - 59T^{2} \)
61 \( 1 + 11.3iT - 61T^{2} \)
67 \( 1 + (-4.17 - 4.17i)T + 67iT^{2} \)
71 \( 1 - 1.94T + 71T^{2} \)
73 \( 1 + (9.10 + 9.10i)T + 73iT^{2} \)
79 \( 1 - 12.3T + 79T^{2} \)
83 \( 1 + (5.04 + 5.04i)T + 83iT^{2} \)
89 \( 1 - 8.87iT - 89T^{2} \)
97 \( 1 + (-5.07 - 5.07i)T + 97iT^{2} \)
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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

−10.04239101660461982018704850261, −9.005198340482888862632792702102, −8.334890717173062834053383640771, −7.72646896678695733559574806631, −7.25985756846638070444232923473, −6.48439034285673158130272013608, −5.15699029153443866974738053534, −3.44652909206137155744186298330, −1.93984552360866815712889291885, −0.27236692183794011117527977129, 1.96588820693278253684997314423, 3.07173111766660655339199027040, 3.92403016789768300400644199125, 4.70927345689028832814831815545, 7.01631162898980056227777322540, 8.085243792561637593009997955913, 8.568953929652983270115729698551, 9.227325856985465681756697240678, 10.01345260439353888839705490860, 10.63746811512298369428207733062

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