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Properties

Label 2-805-115.22-c1-0-28
Degree $2$
Conductor $805$
Sign $0.541 + 0.840i$
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  + (−1.60 − 1.60i)2-s + (0.871 − 0.871i)3-s + 3.15i·4-s + (2.02 − 0.951i)5-s − 2.79·6-s + (−0.707 + 0.707i)7-s + (1.85 − 1.85i)8-s + 1.48i·9-s + (−4.77 − 1.72i)10-s + 5.11i·11-s + (2.75 + 2.75i)12-s + (−1.17 + 1.17i)13-s + 2.27·14-s + (0.933 − 2.59i)15-s + 0.347·16-s + (5.05 − 5.05i)17-s + ⋯
L(s)  = 1  + (−1.13 − 1.13i)2-s + (0.503 − 0.503i)3-s + 1.57i·4-s + (0.904 − 0.425i)5-s − 1.14·6-s + (−0.267 + 0.267i)7-s + (0.656 − 0.656i)8-s + 0.493i·9-s + (−1.51 − 0.544i)10-s + 1.54i·11-s + (0.794 + 0.794i)12-s + (−0.326 + 0.326i)13-s + 0.606·14-s + (0.241 − 0.669i)15-s + 0.0868·16-s + (1.22 − 1.22i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.541 + 0.840i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.541 + 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(805\)    =    \(5 \cdot 7 \cdot 23\)
Sign: $0.541 + 0.840i$
Analytic conductor: \(6.42795\)
Root analytic conductor: \(2.53534\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{805} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 805,\ (\ :1/2),\ 0.541 + 0.840i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.998036 - 0.544403i\)
\(L(\frac12)\) \(\approx\) \(0.998036 - 0.544403i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.02 + 0.951i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
23 \( 1 + (4.48 - 1.70i)T \)
good2 \( 1 + (1.60 + 1.60i)T + 2iT^{2} \)
3 \( 1 + (-0.871 + 0.871i)T - 3iT^{2} \)
11 \( 1 - 5.11iT - 11T^{2} \)
13 \( 1 + (1.17 - 1.17i)T - 13iT^{2} \)
17 \( 1 + (-5.05 + 5.05i)T - 17iT^{2} \)
19 \( 1 + 0.177T + 19T^{2} \)
29 \( 1 - 5.33iT - 29T^{2} \)
31 \( 1 - 7.33T + 31T^{2} \)
37 \( 1 + (-0.899 + 0.899i)T - 37iT^{2} \)
41 \( 1 - 9.00T + 41T^{2} \)
43 \( 1 + (4.95 + 4.95i)T + 43iT^{2} \)
47 \( 1 + (-8.09 - 8.09i)T + 47iT^{2} \)
53 \( 1 + (-6.96 - 6.96i)T + 53iT^{2} \)
59 \( 1 - 0.00127iT - 59T^{2} \)
61 \( 1 + 3.67iT - 61T^{2} \)
67 \( 1 + (-6.28 + 6.28i)T - 67iT^{2} \)
71 \( 1 + 15.2T + 71T^{2} \)
73 \( 1 + (-2.76 + 2.76i)T - 73iT^{2} \)
79 \( 1 + 7.06T + 79T^{2} \)
83 \( 1 + (7.31 + 7.31i)T + 83iT^{2} \)
89 \( 1 + 9.74T + 89T^{2} \)
97 \( 1 + (-3.61 + 3.61i)T - 97iT^{2} \)
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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

−9.922516862207083531822282981337, −9.481241213707345026065624769177, −8.747877843449745906233524137306, −7.74018976163535910197465496673, −7.15503086146352254207730331496, −5.67369286740582972191186329870, −4.58350252590843408691642846686, −2.85593082049306423577526342954, −2.20825973372475686405511866854, −1.28528026053886760469361446703, 0.906117458186559547876489299736, 2.84963257119105158520601278287, 3.89661407519377767035158316052, 5.83811604299168265221174379434, 5.95468766313556809650426304379, 6.97113956794000231147660713080, 8.189813106965599236390013661004, 8.481408812813016682943610778861, 9.556435951991014427800237494115, 10.06889057814623859272284458406

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