Properties

Label 2-845-845.392-c1-0-22
Degree $2$
Conductor $845$
Sign $-0.839 - 0.542i$
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  + (−1.17 + 2.48i)2-s + (1.41 + 1.30i)3-s + (−3.51 − 4.30i)4-s + (−2.13 − 0.659i)5-s + (−4.90 + 1.97i)6-s + (3.90 + 1.13i)7-s + (9.51 − 2.34i)8-s + (0.0560 + 0.693i)9-s + (4.15 − 4.53i)10-s + (−1.39 − 1.63i)11-s + (0.645 − 10.6i)12-s + (3.31 − 1.42i)13-s + (−7.41 + 8.37i)14-s + (−2.15 − 3.71i)15-s + (−3.16 + 15.5i)16-s + (0.355 + 0.645i)17-s + ⋯
L(s)  = 1  + (−0.833 + 1.75i)2-s + (0.815 + 0.752i)3-s + (−1.75 − 2.15i)4-s + (−0.955 − 0.295i)5-s + (−2.00 + 0.805i)6-s + (1.47 + 0.427i)7-s + (3.36 − 0.829i)8-s + (0.0186 + 0.231i)9-s + (1.31 − 1.43i)10-s + (−0.419 − 0.493i)11-s + (0.186 − 3.08i)12-s + (0.918 − 0.396i)13-s + (−1.98 + 2.23i)14-s + (−0.557 − 0.959i)15-s + (−0.791 + 3.87i)16-s + (0.0861 + 0.156i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.839 - 0.542i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.839 - 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.839 - 0.542i$
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: \((2,\ 845,\ (\ :1/2),\ -0.839 - 0.542i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.342274 + 1.16061i\)
\(L(\frac12)\) \(\approx\) \(0.342274 + 1.16061i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.13 + 0.659i)T \)
13 \( 1 + (-3.31 + 1.42i)T \)
good2 \( 1 + (1.17 - 2.48i)T + (-1.26 - 1.54i)T^{2} \)
3 \( 1 + (-1.41 - 1.30i)T + (0.241 + 2.99i)T^{2} \)
7 \( 1 + (-3.90 - 1.13i)T + (5.91 + 3.74i)T^{2} \)
11 \( 1 + (1.39 + 1.63i)T + (-1.76 + 10.8i)T^{2} \)
17 \( 1 + (-0.355 - 0.645i)T + (-9.08 + 14.3i)T^{2} \)
19 \( 1 + (-0.396 + 0.106i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (1.85 - 6.92i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-4.88 - 2.31i)T + (18.3 + 22.4i)T^{2} \)
31 \( 1 + (-3.25 - 2.55i)T + (7.41 + 30.0i)T^{2} \)
37 \( 1 + (-6.19 - 8.24i)T + (-10.2 + 35.5i)T^{2} \)
41 \( 1 + (8.99 + 8.29i)T + (3.29 + 40.8i)T^{2} \)
43 \( 1 + (-0.752 - 5.30i)T + (-41.3 + 11.9i)T^{2} \)
47 \( 1 + (-2.89 + 1.09i)T + (35.1 - 31.1i)T^{2} \)
53 \( 1 + (-2.47 + 4.08i)T + (-24.6 - 46.9i)T^{2} \)
59 \( 1 + (3.77 + 2.49i)T + (23.1 + 54.2i)T^{2} \)
61 \( 1 + (4.50 - 4.69i)T + (-2.45 - 60.9i)T^{2} \)
67 \( 1 + (2.39 - 2.93i)T + (-13.4 - 65.6i)T^{2} \)
71 \( 1 + (-2.07 + 9.21i)T + (-64.1 - 30.4i)T^{2} \)
73 \( 1 + (-6.73 + 9.75i)T + (-25.8 - 68.2i)T^{2} \)
79 \( 1 + (-6.34 + 2.40i)T + (59.1 - 52.3i)T^{2} \)
83 \( 1 + (-2.53 - 4.83i)T + (-47.1 + 68.3i)T^{2} \)
89 \( 1 + (5.24 + 1.40i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (9.76 - 3.26i)T + (77.5 - 58.2i)T^{2} \)
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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

−10.20618908698362050899090165515, −9.167702940119160946067343862666, −8.493481100625056440082178874486, −8.192429955754125528139780014140, −7.59621423915706818526360747267, −6.31062528071229465649429562869, −5.23751101017907785002019132547, −4.64814634799584186037257107729, −3.53662366814483101585925668035, −1.18171828842851079675012404664, 0.948219354654747299346541482073, 2.04108416683554008642250027157, 2.85080940132162093973244755359, 4.10583648256515192454086532454, 4.64930014318936016351185435411, 7.04847799563827931601321955813, 7.955888898769310936863598737451, 8.163139356468164890687912315259, 8.822010987270323253331183820634, 10.08038495050349633461421802221

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