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Properties

Label 2-605-55.49-c1-0-37
Degree $2$
Conductor $605$
Sign $0.718 + 0.695i$
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  + (2.40 − 0.780i)2-s + (−0.465 + 0.640i)3-s + (3.53 − 2.56i)4-s + (2.23 + 0.00508i)5-s + (−0.618 + 1.90i)6-s + (−2.03 − 2.80i)7-s + (3.51 − 4.84i)8-s + (0.733 + 2.25i)9-s + (5.37 − 1.73i)10-s + 3.46i·12-s + (−7.07 − 5.13i)14-s + (−1.04 + 1.43i)15-s + (1.96 − 6.06i)16-s + (1.50 + 0.489i)17-s + (3.51 + 4.84i)18-s + (−3.23 − 2.35i)19-s + ⋯
L(s)  = 1  + (1.69 − 0.551i)2-s + (−0.268 + 0.370i)3-s + (1.76 − 1.28i)4-s + (0.999 + 0.00227i)5-s + (−0.252 + 0.776i)6-s + (−0.769 − 1.05i)7-s + (1.24 − 1.71i)8-s + (0.244 + 0.752i)9-s + (1.69 − 0.547i)10-s + 0.999i·12-s + (−1.89 − 1.37i)14-s + (−0.269 + 0.369i)15-s + (0.492 − 1.51i)16-s + (0.365 + 0.118i)17-s + (0.829 + 1.14i)18-s + (−0.742 − 0.539i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.718 + 0.695i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.718 + 0.695i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.55192 - 1.43861i\)
\(L(\frac12)\) \(\approx\) \(3.55192 - 1.43861i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.23 - 0.00508i)T \)
11 \( 1 \)
good2 \( 1 + (-2.40 + 0.780i)T + (1.61 - 1.17i)T^{2} \)
3 \( 1 + (0.465 - 0.640i)T + (-0.927 - 2.85i)T^{2} \)
7 \( 1 + (2.03 + 2.80i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.50 - 0.489i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (3.23 + 2.35i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 0.792iT - 23T^{2} \)
29 \( 1 + (7.07 - 5.13i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.04 - 3.20i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (0.639 + 0.879i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-7.07 - 5.13i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 3.46iT - 43T^{2} \)
47 \( 1 + (-3.89 + 5.36i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (9.60 - 3.12i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (5.96 - 4.33i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.230 + 0.708i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 9.30iT - 67T^{2} \)
71 \( 1 + (3.12 - 9.62i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (4.07 + 5.60i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-0.387 - 1.19i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (6.30 + 2.04i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + 1.37T + 89T^{2} \)
97 \( 1 + (-5.55 + 1.80i)T + (78.4 - 57.0i)T^{2} \)
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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

−10.59472725696845483071300826328, −10.26017343504423788206146933627, −9.201494161280566043576924615971, −7.40437325688863472094044676538, −6.54795146403858567034069945420, −5.69977368738169796290709704705, −4.87086072848572162455075308331, −4.00469798747666858257091394042, −2.94677705504163945841723603534, −1.69491319047124218028586874241, 2.12093176153469214126668260302, 3.15280976114476202187414586451, 4.31127079393191198449975254747, 5.66782570188004116308141187233, 5.97349051945497336305407498647, 6.60683788236493246148096315597, 7.66687565223987265251883417365, 9.085705041203459123079616252366, 9.822029283027394111630386724485, 11.17356216523370135777568894380

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