Properties

Label 2-845-845.392-c1-0-23
Degree $2$
Conductor $845$
Sign $0.411 - 0.911i$
Analytic cond. $6.74735$
Root an. cond. $2.59756$
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  + (−0.290 + 0.613i)2-s + (−0.720 − 0.664i)3-s + (0.973 + 1.19i)4-s + (2.08 + 0.816i)5-s + (0.616 − 0.248i)6-s + (−2.00 − 0.580i)7-s + (−2.33 + 0.574i)8-s + (−0.164 − 2.03i)9-s + (−1.10 + 1.03i)10-s + (2.18 + 2.57i)11-s + (0.0910 − 1.50i)12-s + (0.949 − 3.47i)13-s + (0.939 − 1.06i)14-s + (−0.956 − 1.97i)15-s + (−0.289 + 1.41i)16-s + (3.58 + 6.50i)17-s + ⋯
L(s)  = 1  + (−0.205 + 0.433i)2-s + (−0.415 − 0.383i)3-s + (0.486 + 0.596i)4-s + (0.930 + 0.365i)5-s + (0.251 − 0.101i)6-s + (−0.757 − 0.219i)7-s + (−0.824 + 0.203i)8-s + (−0.0547 − 0.677i)9-s + (−0.349 + 0.328i)10-s + (0.659 + 0.775i)11-s + (0.0262 − 0.434i)12-s + (0.263 − 0.964i)13-s + (0.251 − 0.283i)14-s + (−0.246 − 0.508i)15-s + (−0.0724 + 0.354i)16-s + (0.868 + 1.57i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.411 - 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.411 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.411 - 0.911i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (392, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.411 - 0.911i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28749 + 0.831078i\)
\(L(\frac12)\) \(\approx\) \(1.28749 + 0.831078i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.08 - 0.816i)T \)
13 \( 1 + (-0.949 + 3.47i)T \)
good2 \( 1 + (0.290 - 0.613i)T + (-1.26 - 1.54i)T^{2} \)
3 \( 1 + (0.720 + 0.664i)T + (0.241 + 2.99i)T^{2} \)
7 \( 1 + (2.00 + 0.580i)T + (5.91 + 3.74i)T^{2} \)
11 \( 1 + (-2.18 - 2.57i)T + (-1.76 + 10.8i)T^{2} \)
17 \( 1 + (-3.58 - 6.50i)T + (-9.08 + 14.3i)T^{2} \)
19 \( 1 + (-4.53 + 1.21i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.0674 + 0.251i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (0.133 + 0.0631i)T + (18.3 + 22.4i)T^{2} \)
31 \( 1 + (-2.36 - 1.85i)T + (7.41 + 30.0i)T^{2} \)
37 \( 1 + (0.737 + 0.981i)T + (-10.2 + 35.5i)T^{2} \)
41 \( 1 + (-4.70 - 4.34i)T + (3.29 + 40.8i)T^{2} \)
43 \( 1 + (-0.657 - 4.63i)T + (-41.3 + 11.9i)T^{2} \)
47 \( 1 + (8.45 - 3.20i)T + (35.1 - 31.1i)T^{2} \)
53 \( 1 + (-4.52 + 7.47i)T + (-24.6 - 46.9i)T^{2} \)
59 \( 1 + (-9.42 - 6.22i)T + (23.1 + 54.2i)T^{2} \)
61 \( 1 + (-1.70 + 1.77i)T + (-2.45 - 60.9i)T^{2} \)
67 \( 1 + (1.75 - 2.15i)T + (-13.4 - 65.6i)T^{2} \)
71 \( 1 + (0.527 - 2.34i)T + (-64.1 - 30.4i)T^{2} \)
73 \( 1 + (2.22 - 3.22i)T + (-25.8 - 68.2i)T^{2} \)
79 \( 1 + (6.78 - 2.57i)T + (59.1 - 52.3i)T^{2} \)
83 \( 1 + (-0.892 - 1.69i)T + (-47.1 + 68.3i)T^{2} \)
89 \( 1 + (-6.24 - 1.67i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-15.1 + 5.04i)T + (77.5 - 58.2i)T^{2} \)
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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

−10.11971710967429238347022628287, −9.634856647328390634407428276748, −8.586773552626300404505715425438, −7.56953261360929275860872054001, −6.75013636880095632246383603960, −6.24515205218433590150179026142, −5.53978809079761172616284601320, −3.70245287174061571079254697267, −2.93680779439964822564658001173, −1.37010134927056824541104926241, 0.954390512720863159499241398719, 2.26122595157621421083174427134, 3.34955528059901373366836472428, 4.93429027319567478225869528075, 5.68905723739349196112446855152, 6.30501511358803348288923087958, 7.29346404397513962250557250307, 8.796047889131187784338828224716, 9.547687871002729353583524216959, 9.881817919357954577254865290254

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