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Properties

Label 2-805-7.2-c1-0-5
Degree $2$
Conductor $805$
Sign $0.666 - 0.745i$
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  + (0.152 − 0.263i)2-s + (−1.43 − 2.49i)3-s + (0.953 + 1.65i)4-s + (0.5 − 0.866i)5-s − 0.875·6-s + (−2.56 + 0.666i)7-s + 1.18·8-s + (−2.64 + 4.58i)9-s + (−0.152 − 0.263i)10-s + (2.27 + 3.93i)11-s + (2.74 − 4.75i)12-s − 2.49·13-s + (−0.213 + 0.775i)14-s − 2.87·15-s + (−1.72 + 2.99i)16-s + (0.734 + 1.27i)17-s + ⋯
L(s)  = 1  + (0.107 − 0.186i)2-s + (−0.831 − 1.43i)3-s + (0.476 + 0.825i)4-s + (0.223 − 0.387i)5-s − 0.357·6-s + (−0.967 + 0.252i)7-s + 0.420·8-s + (−0.881 + 1.52i)9-s + (−0.0480 − 0.0833i)10-s + (0.684 + 1.18i)11-s + (0.792 − 1.37i)12-s − 0.691·13-s + (−0.0571 + 0.207i)14-s − 0.743·15-s + (−0.431 + 0.747i)16-s + (0.178 + 0.308i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.666 - 0.745i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(805\)    =    \(5 \cdot 7 \cdot 23\)
Sign: $0.666 - 0.745i$
Analytic conductor: \(6.42795\)
Root analytic conductor: \(2.53534\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{805} (576, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 805,\ (\ :1/2),\ 0.666 - 0.745i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.812005 + 0.363480i\)
\(L(\frac12)\) \(\approx\) \(0.812005 + 0.363480i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (2.56 - 0.666i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.152 + 0.263i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (1.43 + 2.49i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (-2.27 - 3.93i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.49T + 13T^{2} \)
17 \( 1 + (-0.734 - 1.27i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.63 - 4.56i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 - 2.82T + 29T^{2} \)
31 \( 1 + (-0.218 - 0.378i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.09 - 7.08i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.27T + 41T^{2} \)
43 \( 1 + 1.00T + 43T^{2} \)
47 \( 1 + (0.0459 - 0.0795i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.12 + 3.68i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.142 - 0.246i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.36 + 9.29i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.85 - 11.8i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.35T + 71T^{2} \)
73 \( 1 + (1.33 + 2.31i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.06 + 1.84i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.33T + 83T^{2} \)
89 \( 1 + (8.56 - 14.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 16.6T + 97T^{2} \)
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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

−10.46012115212467795979472630386, −9.627318354842804647927612972622, −8.392415797709717656609306710449, −7.61801908615671440071583911243, −6.70180173348055865932997477289, −6.43023559844834820897999609828, −5.20362355623629447583342083321, −3.89317726201762547386671934169, −2.45206967812658586099438193725, −1.56005968257561676508319291035, 0.46129626329342257267384093550, 2.74745368870574583427741621084, 3.86102721769452041236246887948, 4.88727718709823570453344091595, 5.78075625989988309688023319184, 6.36916637559100194215452346213, 7.13164602191617799067745503569, 8.909076133350222058312369122593, 9.569562654256628750740102080104, 10.22517050360894020058365381977

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