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Properties

Label 2-605-55.43-c1-0-42
Degree $2$
Conductor $605$
Sign $-0.625 + 0.780i$
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  + (0.407 + 0.407i)2-s + (−0.544 − 0.544i)3-s − 1.66i·4-s + (0.752 − 2.10i)5-s − 0.443i·6-s + (−0.843 − 0.843i)7-s + (1.49 − 1.49i)8-s − 2.40i·9-s + (1.16 − 0.550i)10-s + (−0.908 + 0.908i)12-s + (−4.29 + 4.29i)13-s − 0.687i·14-s + (−1.55 + 0.736i)15-s − 2.12·16-s + (0.262 + 0.262i)17-s + (0.980 − 0.980i)18-s + ⋯
L(s)  = 1  + (0.287 + 0.287i)2-s + (−0.314 − 0.314i)3-s − 0.834i·4-s + (0.336 − 0.941i)5-s − 0.180i·6-s + (−0.318 − 0.318i)7-s + (0.528 − 0.528i)8-s − 0.802i·9-s + (0.368 − 0.174i)10-s + (−0.262 + 0.262i)12-s + (−1.19 + 1.19i)13-s − 0.183i·14-s + (−0.401 + 0.190i)15-s − 0.530·16-s + (0.0635 + 0.0635i)17-s + (0.231 − 0.231i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.625 + 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.625 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.625 + 0.780i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (483, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.625 + 0.780i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.555589 - 1.15797i\)
\(L(\frac12)\) \(\approx\) \(0.555589 - 1.15797i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.752 + 2.10i)T \)
11 \( 1 \)
good2 \( 1 + (-0.407 - 0.407i)T + 2iT^{2} \)
3 \( 1 + (0.544 + 0.544i)T + 3iT^{2} \)
7 \( 1 + (0.843 + 0.843i)T + 7iT^{2} \)
13 \( 1 + (4.29 - 4.29i)T - 13iT^{2} \)
17 \( 1 + (-0.262 - 0.262i)T + 17iT^{2} \)
19 \( 1 + 1.37T + 19T^{2} \)
23 \( 1 + (-3.48 - 3.48i)T + 23iT^{2} \)
29 \( 1 - 6.12T + 29T^{2} \)
31 \( 1 + 2.47T + 31T^{2} \)
37 \( 1 + (-2.10 + 2.10i)T - 37iT^{2} \)
41 \( 1 + 6.12iT - 41T^{2} \)
43 \( 1 + (-6.75 + 6.75i)T - 43iT^{2} \)
47 \( 1 + (-0.902 + 0.902i)T - 47iT^{2} \)
53 \( 1 + (0.288 + 0.288i)T + 53iT^{2} \)
59 \( 1 + 0.187iT - 59T^{2} \)
61 \( 1 + 0.683iT - 61T^{2} \)
67 \( 1 + (-7.14 + 7.14i)T - 67iT^{2} \)
71 \( 1 - 1.03T + 71T^{2} \)
73 \( 1 + (3.23 - 3.23i)T - 73iT^{2} \)
79 \( 1 - 11.6T + 79T^{2} \)
83 \( 1 + (8.04 - 8.04i)T - 83iT^{2} \)
89 \( 1 + 8.04iT - 89T^{2} \)
97 \( 1 + (-5.96 + 5.96i)T - 97iT^{2} \)
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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

−10.15085769415180921416210947553, −9.465919685817837433572579086213, −8.872363161086297128288253435501, −7.29288230381315108673852495703, −6.68035863088411916722112241485, −5.74801600502580696886095131207, −4.91455049803033185742978858312, −3.98110957143292790659284014977, −1.97527352469052198442542205675, −0.66162632049204445559939320588, 2.51144354728833441018164855180, 2.96576440701077996124161656192, 4.43816014225890584601027186636, 5.31129640882213266104320452918, 6.46165102732003032758437458583, 7.50352985306094507773242743078, 8.128423242505893270072388288905, 9.425993088818079950690068004238, 10.36719697128033811624919700306, 10.87093007383177852392761703781

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