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Properties

Label 2-805-805.114-c0-0-1
Degree $2$
Conductor $805$
Sign $0.0633 + 0.997i$
Analytic cond. $0.401747$
Root an. cond. $0.633835$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)5-s + (0.5 − 0.866i)7-s + (−0.5 + 0.866i)9-s + (−0.499 + 0.866i)16-s + (−0.5 − 0.866i)17-s − 0.999·20-s + (0.5 − 0.866i)23-s + (−0.499 − 0.866i)25-s − 0.999·28-s − 29-s + (0.5 + 0.866i)31-s + (−0.499 − 0.866i)35-s + 0.999·36-s + (−0.5 + 0.866i)37-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)5-s + (0.5 − 0.866i)7-s + (−0.5 + 0.866i)9-s + (−0.499 + 0.866i)16-s + (−0.5 − 0.866i)17-s − 0.999·20-s + (0.5 − 0.866i)23-s + (−0.499 − 0.866i)25-s − 0.999·28-s − 29-s + (0.5 + 0.866i)31-s + (−0.499 − 0.866i)35-s + 0.999·36-s + (−0.5 + 0.866i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(805\)    =    \(5 \cdot 7 \cdot 23\)
Sign: $0.0633 + 0.997i$
Analytic conductor: \(0.401747\)
Root analytic conductor: \(0.633835\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{805} (114, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 805,\ (\ :0),\ 0.0633 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9201386186\)
\(L(\frac12)\) \(\approx\) \(0.9201386186\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (0.5 + 0.866i)T^{2} \)
3 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - 2T + T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−10.33344898494582588375374667991, −9.359526772370627386776936703247, −8.720616205736558959617197683903, −7.84529941717167972719610223389, −6.71695476388152319705518368387, −5.58053920085823756834659580132, −4.92141652423086141310081278189, −4.26294965423188492473124359943, −2.34648688998682591093358843194, −1.03570739654424254882588450903, 2.20143225728267599686096570944, 3.20378970900147166258128068974, 4.15310648763291175333065615763, 5.53643597884971034837514250403, 6.20669076629945210459308373751, 7.34435381266134988689358655165, 8.125007468632201676129809676782, 9.246070253599866680918870600771, 9.355844557802794841477617471135, 10.90238051366225894486611533304

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