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Properties

Label 2-605-55.17-c1-0-41
Degree $2$
Conductor $605$
Sign $-0.0135 + 0.999i$
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  + (3.01 − 0.477i)3-s + (−1.17 − 1.61i)4-s + (−0.914 − 2.04i)5-s + (6.00 − 1.95i)9-s + (−4.31 − 4.31i)12-s + (−3.73 − 5.71i)15-s + (−1.23 + 3.80i)16-s + (−2.22 + 3.87i)20-s + (2.84 − 2.84i)23-s + (−3.32 + 3.73i)25-s + (9.01 − 4.59i)27-s + (−3.07 − 9.46i)31-s + (−10.2 − 7.42i)36-s + (−2.06 − 0.326i)37-s + (−9.47 − 10.4i)45-s + ⋯
L(s)  = 1  + (1.74 − 0.275i)3-s + (−0.587 − 0.809i)4-s + (−0.408 − 0.912i)5-s + (2.00 − 0.650i)9-s + (−1.24 − 1.24i)12-s + (−0.963 − 1.47i)15-s + (−0.309 + 0.951i)16-s + (−0.498 + 0.867i)20-s + (0.592 − 0.592i)23-s + (−0.665 + 0.746i)25-s + (1.73 − 0.884i)27-s + (−0.552 − 1.69i)31-s + (−1.70 − 1.23i)36-s + (−0.338 − 0.0536i)37-s + (−1.41 − 1.56i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0135 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0135 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.0135 + 0.999i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.0135 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48816 - 1.50840i\)
\(L(\frac12)\) \(\approx\) \(1.48816 - 1.50840i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.914 + 2.04i)T \)
11 \( 1 \)
good2 \( 1 + (1.17 + 1.61i)T^{2} \)
3 \( 1 + (-3.01 + 0.477i)T + (2.85 - 0.927i)T^{2} \)
7 \( 1 + (-6.65 - 2.16i)T^{2} \)
13 \( 1 + (-7.64 - 10.5i)T^{2} \)
17 \( 1 + (-9.99 + 13.7i)T^{2} \)
19 \( 1 + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-2.84 + 2.84i)T - 23iT^{2} \)
29 \( 1 + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (3.07 + 9.46i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (2.06 + 0.326i)T + (35.1 + 11.4i)T^{2} \)
41 \( 1 + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 + (-2.06 - 13.0i)T + (-44.6 + 14.5i)T^{2} \)
53 \( 1 + (-4.57 - 2.33i)T + (31.1 + 42.8i)T^{2} \)
59 \( 1 + (-1.94 - 2.68i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (49.3 + 35.8i)T^{2} \)
67 \( 1 + (-11.4 - 11.4i)T + 67iT^{2} \)
71 \( 1 + (0.927 - 2.85i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (48.7 - 67.1i)T^{2} \)
89 \( 1 - 9iT - 89T^{2} \)
97 \( 1 + (-2.26 + 4.44i)T + (-57.0 - 78.4i)T^{2} \)
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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

−10.02605348309719501411990715030, −9.288385760927526025458498642521, −8.787972557378268197451744357851, −8.053433540414615973572651614585, −7.17956848519049418518840029320, −5.77579155370593698166748477970, −4.53823465708622212259629291495, −3.83490600216394763651706411216, −2.37844675308655739434825305894, −1.07506306808964146299660533999, 2.28931363387556777297338639608, 3.38283114039886425271348445448, 3.71704103840765248576662869448, 4.97976626413970239670850356147, 6.94755787146078198580013921757, 7.47372572382281175054063805207, 8.388187175701340257198421151057, 8.897698653915576352787071473650, 9.810002364256281300782831737471, 10.62674812401904814306129631348

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