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Properties

Label 2-805-7.2-c1-0-31
Degree $2$
Conductor $805$
Sign $0.910 - 0.413i$
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  + (−1.27 + 2.21i)2-s + (0.868 + 1.50i)3-s + (−2.26 − 3.92i)4-s + (0.5 − 0.866i)5-s − 4.43·6-s + (−2.61 − 0.422i)7-s + 6.47·8-s + (−0.00851 + 0.0147i)9-s + (1.27 + 2.21i)10-s + (0.671 + 1.16i)11-s + (3.93 − 6.82i)12-s − 5.53·13-s + (4.27 − 5.24i)14-s + 1.73·15-s + (−3.74 + 6.48i)16-s + (−2.07 − 3.58i)17-s + ⋯
L(s)  = 1  + (−0.903 + 1.56i)2-s + (0.501 + 0.868i)3-s + (−1.13 − 1.96i)4-s + (0.223 − 0.387i)5-s − 1.81·6-s + (−0.987 − 0.159i)7-s + 2.29·8-s + (−0.00283 + 0.00491i)9-s + (0.404 + 0.700i)10-s + (0.202 + 0.350i)11-s + (1.13 − 1.96i)12-s − 1.53·13-s + (1.14 − 1.40i)14-s + 0.448·15-s + (−0.936 + 1.62i)16-s + (−0.502 − 0.870i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.413i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 - 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(805\)    =    \(5 \cdot 7 \cdot 23\)
Sign: $0.910 - 0.413i$
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: \((2,\ 805,\ (\ :1/2),\ 0.910 - 0.413i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.632888 + 0.137019i\)
\(L(\frac12)\) \(\approx\) \(0.632888 + 0.137019i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.61 + 0.422i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (1.27 - 2.21i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.868 - 1.50i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (-0.671 - 1.16i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.53T + 13T^{2} \)
17 \( 1 + (2.07 + 3.58i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.86 + 4.96i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 - 6.19T + 29T^{2} \)
31 \( 1 + (4.30 + 7.44i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.42 - 4.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 8.82T + 41T^{2} \)
43 \( 1 - 8.80T + 43T^{2} \)
47 \( 1 + (-3.33 + 5.77i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.75 + 6.49i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.88 + 11.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.14 - 8.90i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.50 + 7.80i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.6T + 71T^{2} \)
73 \( 1 + (0.907 + 1.57i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.50 + 2.60i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 13.8T + 83T^{2} \)
89 \( 1 + (-4.69 + 8.13i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 13.9T + 97T^{2} \)
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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

−9.706682284911768637605806470816, −9.361539225749170005881086288636, −8.930087347879013439816468879700, −7.58940536687509138127101304551, −7.07434881279666221650802793487, −6.17355484324989298670573882364, −5.02509394763812294277335326038, −4.39918037823520523084557228373, −2.77037648355353336899809389961, −0.43131533223228949374341945527, 1.35482274674433314534816060441, 2.47757090525876370112525734017, 3.03335562757872904588847080444, 4.25180159504111791940078425348, 5.97896901639941043767786728591, 7.19369539462693365118473527003, 7.77837778333301464920946811588, 8.832465034581870739197203652542, 9.400134433887191113781068794038, 10.36570193685982298838273421451

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