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Properties

Label 2-605-5.4-c1-0-39
Degree $2$
Conductor $605$
Sign $0.131 + 0.991i$
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  + 2.02i·2-s − 2.62i·3-s − 2.09·4-s + (−0.294 − 2.21i)5-s + 5.30·6-s − 0.965i·7-s − 0.192i·8-s − 3.86·9-s + (4.48 − 0.596i)10-s + 5.49i·12-s − 4.52i·13-s + 1.95·14-s + (−5.80 + 0.772i)15-s − 3.80·16-s + 3.33i·17-s − 7.82i·18-s + ⋯
L(s)  = 1  + 1.43i·2-s − 1.51i·3-s − 1.04·4-s + (−0.131 − 0.991i)5-s + 2.16·6-s − 0.365i·7-s − 0.0681i·8-s − 1.28·9-s + (1.41 − 0.188i)10-s + 1.58i·12-s − 1.25i·13-s + 0.522·14-s + (−1.49 + 0.199i)15-s − 0.950·16-s + 0.809i·17-s − 1.84i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.131 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.131 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.131 + 0.991i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (364, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.131 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.784594 - 0.687102i\)
\(L(\frac12)\) \(\approx\) \(0.784594 - 0.687102i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.294 + 2.21i)T \)
11 \( 1 \)
good2 \( 1 - 2.02iT - 2T^{2} \)
3 \( 1 + 2.62iT - 3T^{2} \)
7 \( 1 + 0.965iT - 7T^{2} \)
13 \( 1 + 4.52iT - 13T^{2} \)
17 \( 1 - 3.33iT - 17T^{2} \)
19 \( 1 + 3.27T + 19T^{2} \)
23 \( 1 + 3.36iT - 23T^{2} \)
29 \( 1 + 4.91T + 29T^{2} \)
31 \( 1 + 0.418T + 31T^{2} \)
37 \( 1 + 6.33iT - 37T^{2} \)
41 \( 1 - 5.78T + 41T^{2} \)
43 \( 1 + 2.26iT - 43T^{2} \)
47 \( 1 + 4.32iT - 47T^{2} \)
53 \( 1 + 2.66iT - 53T^{2} \)
59 \( 1 - 10.1T + 59T^{2} \)
61 \( 1 + 2.47T + 61T^{2} \)
67 \( 1 - 9.60iT - 67T^{2} \)
71 \( 1 + 5.45T + 71T^{2} \)
73 \( 1 - 1.43iT - 73T^{2} \)
79 \( 1 + 1.00T + 79T^{2} \)
83 \( 1 - 7.39iT - 83T^{2} \)
89 \( 1 - 12.1T + 89T^{2} \)
97 \( 1 - 3.01iT - 97T^{2} \)
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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

−10.41391675028223855626961219574, −8.926509137058294112300276198387, −8.302788728935106550350850435635, −7.69821375626186457313113199362, −7.01751349203118027310215843188, −6.00792426558735708445489853389, −5.45640998906019935170527920094, −4.12730150706244734478417478666, −2.12908802462747729473994127798, −0.56140007263244585678820173160, 2.14798618954291714549092463003, 3.17312691171254883201512406894, 3.97536409122043858520525995727, 4.76607049985027916624379858828, 6.16262312680056186719978525559, 7.32672446591276216843290003142, 8.923408614089669446481411262192, 9.459184828530090731944750179700, 10.13146153538143561504246394838, 10.89207079527307104735543725862

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