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Properties

Label 2-605-55.28-c1-0-36
Degree $2$
Conductor $605$
Sign $-0.543 + 0.839i$
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  + (0.474 − 0.930i)2-s + (−0.440 − 2.78i)3-s + (0.533 + 0.734i)4-s + (2.23 − 0.0540i)5-s + (−2.79 − 0.909i)6-s + (0.543 + 0.0860i)7-s + (3.00 − 0.475i)8-s + (−4.68 + 1.52i)9-s + (1.01 − 2.10i)10-s + (1.80 − 1.80i)12-s + (−2.89 − 1.47i)13-s + (0.337 − 0.464i)14-s + (−1.13 − 6.19i)15-s + (0.419 − 1.29i)16-s + (5.04 − 2.57i)17-s + (−0.805 + 5.08i)18-s + ⋯
L(s)  = 1  + (0.335 − 0.658i)2-s + (−0.254 − 1.60i)3-s + (0.266 + 0.367i)4-s + (0.999 − 0.0241i)5-s + (−1.14 − 0.371i)6-s + (0.205 + 0.0325i)7-s + (1.06 − 0.168i)8-s + (−1.56 + 0.507i)9-s + (0.319 − 0.666i)10-s + (0.522 − 0.522i)12-s + (−0.801 − 0.408i)13-s + (0.0902 − 0.124i)14-s + (−0.293 − 1.59i)15-s + (0.104 − 0.322i)16-s + (1.22 − 0.623i)17-s + (−0.189 + 1.19i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.543 + 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.543 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.543 + 0.839i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (578, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.543 + 0.839i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02159 - 1.87975i\)
\(L(\frac12)\) \(\approx\) \(1.02159 - 1.87975i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.23 + 0.0540i)T \)
11 \( 1 \)
good2 \( 1 + (-0.474 + 0.930i)T + (-1.17 - 1.61i)T^{2} \)
3 \( 1 + (0.440 + 2.78i)T + (-2.85 + 0.927i)T^{2} \)
7 \( 1 + (-0.543 - 0.0860i)T + (6.65 + 2.16i)T^{2} \)
13 \( 1 + (2.89 + 1.47i)T + (7.64 + 10.5i)T^{2} \)
17 \( 1 + (-5.04 + 2.57i)T + (9.99 - 13.7i)T^{2} \)
19 \( 1 + (1.25 + 0.914i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (0.803 + 0.803i)T + 23iT^{2} \)
29 \( 1 + (3.44 - 2.50i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.509 + 1.56i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (0.149 - 0.945i)T + (-35.1 - 11.4i)T^{2} \)
41 \( 1 + (5.25 - 7.23i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + (2.55 - 2.55i)T - 43iT^{2} \)
47 \( 1 + (-4.02 + 0.636i)T + (44.6 - 14.5i)T^{2} \)
53 \( 1 + (3.19 - 6.27i)T + (-31.1 - 42.8i)T^{2} \)
59 \( 1 + (-3.97 - 5.47i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-8.75 - 2.84i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + (2.62 - 2.62i)T - 67iT^{2} \)
71 \( 1 + (-2.11 + 6.51i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (1.57 - 9.96i)T + (-69.4 - 22.5i)T^{2} \)
79 \( 1 + (1.28 + 3.96i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (5.14 + 10.0i)T + (-48.7 + 67.1i)T^{2} \)
89 \( 1 - 3.64iT - 89T^{2} \)
97 \( 1 + (-14.8 - 7.56i)T + (57.0 + 78.4i)T^{2} \)
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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

−10.54502325764258829613052550239, −9.691998888216842577451254319439, −8.337573211843210851148353705108, −7.51611730596691757391126100838, −6.88467668365010676087806446728, −5.86271668279248868417245503450, −4.91091866583081581341283557503, −3.07518532765838865885579361553, −2.22326609584815124405394952797, −1.24153966116582633177638065453, 1.96452454525829397832231122293, 3.62260122755877910421228400731, 4.78494583717258489841810956360, 5.39828571851602743505636610255, 6.06690866655325442931254602163, 7.17058225913708139552853111573, 8.430582377515846886697526106442, 9.620302849628955916262730252676, 10.00153079436183451790076700623, 10.66244582113942904955961424853

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