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Properties

Label 2-605-55.2-c1-0-31
Degree $2$
Conductor $605$
Sign $0.900 + 0.435i$
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  + (−0.256 + 1.61i)3-s + (1.17 − 1.61i)4-s + (0.914 − 2.04i)5-s + (0.301 + 0.0978i)9-s + (2.31 + 2.31i)12-s + (3.06 + 2.00i)15-s + (−1.23 − 3.80i)16-s + (−2.22 − 3.87i)20-s + (6.15 − 6.15i)23-s + (−3.32 − 3.73i)25-s + (−2.46 + 4.84i)27-s + (3.07 − 9.46i)31-s + (0.512 − 0.372i)36-s + (1.87 + 11.8i)37-s + (0.474 − 0.525i)45-s + ⋯
L(s)  = 1  + (−0.147 + 0.934i)3-s + (0.587 − 0.809i)4-s + (0.408 − 0.912i)5-s + (0.100 + 0.0326i)9-s + (0.668 + 0.668i)12-s + (0.792 + 0.516i)15-s + (−0.309 − 0.951i)16-s + (−0.498 − 0.867i)20-s + (1.28 − 1.28i)23-s + (−0.665 − 0.746i)25-s + (−0.474 + 0.931i)27-s + (0.552 − 1.69i)31-s + (0.0853 − 0.0620i)36-s + (0.308 + 1.94i)37-s + (0.0707 − 0.0782i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.900 + 0.435i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.900 + 0.435i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.72433 - 0.394784i\)
\(L(\frac12)\) \(\approx\) \(1.72433 - 0.394784i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.256 - 1.61i)T + (-2.85 - 0.927i)T^{2} \)
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 \( 1 + (-6.15 + 6.15i)T - 23iT^{2} \)
29 \( 1 + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-3.07 + 9.46i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.87 - 11.8i)T + (-35.1 + 11.4i)T^{2} \)
41 \( 1 + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 + (-3.74 - 0.593i)T + (44.6 + 14.5i)T^{2} \)
53 \( 1 + (-6.18 - 12.1i)T + (-31.1 + 42.8i)T^{2} \)
59 \( 1 + (-1.94 + 2.68i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (49.3 - 35.8i)T^{2} \)
67 \( 1 + (-1.52 - 1.52i)T + 67iT^{2} \)
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 \( 1 + (16.9 - 8.65i)T + (57.0 - 78.4i)T^{2} \)
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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

−10.48330553669475254752454034941, −9.762195751542785009158580401613, −9.195400086146123405279826185958, −8.088376641625526454418287543799, −6.80204214223206900430445294051, −5.87423694076366109508141045408, −4.96061666468802474963467729048, −4.33440839733639375163622078328, −2.60859697832158391955214160796, −1.13197742571490040525759979815, 1.64858304961719970973598643501, 2.76416453178298848091557940747, 3.76851395337762085199032207489, 5.47854949918991475844711649480, 6.60859766778128599120565448925, 7.06822166298138040008802353142, 7.71335865441823603807132871692, 8.838858976293497231274690957450, 9.956398554886401019841737634129, 10.95066744760732180993683267978

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