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Properties

Label 2-805-5.4-c1-0-40
Degree $2$
Conductor $805$
Sign $0.927 + 0.374i$
Analytic cond. $6.42795$
Root an. cond. $2.53534$
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  + 0.157i·2-s + 0.829i·3-s + 1.97·4-s + (−2.07 − 0.837i)5-s − 0.130·6-s − i·7-s + 0.626i·8-s + 2.31·9-s + (0.131 − 0.326i)10-s − 3.37·11-s + 1.63i·12-s − 5.51i·13-s + 0.157·14-s + (0.694 − 1.71i)15-s + 3.85·16-s − 3.82i·17-s + ⋯
L(s)  = 1  + 0.111i·2-s + 0.478i·3-s + 0.987·4-s + (−0.927 − 0.374i)5-s − 0.0533·6-s − 0.377i·7-s + 0.221i·8-s + 0.770·9-s + (0.0417 − 0.103i)10-s − 1.01·11-s + 0.472i·12-s − 1.53i·13-s + 0.0421·14-s + (0.179 − 0.444i)15-s + 0.962·16-s − 0.928i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(805\)    =    \(5 \cdot 7 \cdot 23\)
Sign: $0.927 + 0.374i$
Analytic conductor: \(6.42795\)
Root analytic conductor: \(2.53534\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{805} (484, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 805,\ (\ :1/2),\ 0.927 + 0.374i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67216 - 0.324853i\)
\(L(\frac12)\) \(\approx\) \(1.67216 - 0.324853i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.07 + 0.837i)T \)
7 \( 1 + iT \)
23 \( 1 - iT \)
good2 \( 1 - 0.157iT - 2T^{2} \)
3 \( 1 - 0.829iT - 3T^{2} \)
11 \( 1 + 3.37T + 11T^{2} \)
13 \( 1 + 5.51iT - 13T^{2} \)
17 \( 1 + 3.82iT - 17T^{2} \)
19 \( 1 - 8.00T + 19T^{2} \)
29 \( 1 - 1.05T + 29T^{2} \)
31 \( 1 - 0.791T + 31T^{2} \)
37 \( 1 + 8.93iT - 37T^{2} \)
41 \( 1 + 1.78T + 41T^{2} \)
43 \( 1 - 7.33iT - 43T^{2} \)
47 \( 1 + 7.90iT - 47T^{2} \)
53 \( 1 - 1.75iT - 53T^{2} \)
59 \( 1 - 1.61T + 59T^{2} \)
61 \( 1 + 4.58T + 61T^{2} \)
67 \( 1 - 5.76iT - 67T^{2} \)
71 \( 1 - 16.3T + 71T^{2} \)
73 \( 1 - 3.89iT - 73T^{2} \)
79 \( 1 + 13.4T + 79T^{2} \)
83 \( 1 - 2.22iT - 83T^{2} \)
89 \( 1 - 1.02T + 89T^{2} \)
97 \( 1 - 0.149iT - 97T^{2} \)
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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

−10.28560060765505204704379835398, −9.587884145057466867386271272546, −8.205645937859263746415067550674, −7.50385083121366653543093223976, −7.16258104277357293179522096021, −5.54351458341071141589574920718, −4.97199750542161341484232323903, −3.61643370064276382315711445400, −2.84024711069035793596871748747, −0.935136640801432098520345851764, 1.47486294841242646777086412074, 2.63919761108261397323599226277, 3.69960736154139734675439754156, 4.90516020350389619483209665527, 6.24318327092212392134216722277, 6.95800532031559201367062760209, 7.60189893494166443297654164714, 8.300176375849741599928453135847, 9.633370827995553792665451548512, 10.45292018943335916781840214951

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