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Properties

Label 2-845-13.4-c1-0-10
Degree $2$
Conductor $845$
Sign $-0.0771 - 0.997i$
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  + (−0.358 − 0.207i)2-s + (−0.707 + 1.22i)3-s + (−0.914 − 1.58i)4-s + i·5-s + (0.507 − 0.292i)6-s + (0.717 − 0.414i)7-s + 1.58i·8-s + (0.500 + 0.866i)9-s + (0.207 − 0.358i)10-s + (0.507 + 0.292i)11-s + 2.58·12-s − 0.343·14-s + (−1.22 − 0.707i)15-s + (−1.49 + 2.59i)16-s + (−2.41 − 4.18i)17-s − 0.414i·18-s + ⋯
L(s)  = 1  + (−0.253 − 0.146i)2-s + (−0.408 + 0.707i)3-s + (−0.457 − 0.791i)4-s + 0.447i·5-s + (0.207 − 0.119i)6-s + (0.271 − 0.156i)7-s + 0.560i·8-s + (0.166 + 0.288i)9-s + (0.0654 − 0.113i)10-s + (0.152 + 0.0883i)11-s + 0.746·12-s − 0.0917·14-s + (−0.316 − 0.182i)15-s + (−0.374 + 0.649i)16-s + (−0.585 − 1.01i)17-s − 0.0976i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0771 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0771 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.0771 - 0.997i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.0771 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.559619 + 0.604596i\)
\(L(\frac12)\) \(\approx\) \(0.559619 + 0.604596i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - iT \)
13 \( 1 \)
good2 \( 1 + (0.358 + 0.207i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (0.707 - 1.22i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (-0.717 + 0.414i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.507 - 0.292i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.41 + 4.18i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.95 + 1.70i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.707 - 1.22i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.82 - 4.89i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 10.2iT - 31T^{2} \)
37 \( 1 + (-7.34 - 4.24i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-7.64 - 4.41i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.53 - 2.65i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 0.828iT - 47T^{2} \)
53 \( 1 + 14.4T + 53T^{2} \)
59 \( 1 + (8.87 - 5.12i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4 - 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.73 - i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.84 - 3.94i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 8.48iT - 73T^{2} \)
79 \( 1 - 8.48T + 79T^{2} \)
83 \( 1 + 8.82iT - 83T^{2} \)
89 \( 1 + (-5.19 - 3i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.16 + 1.82i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.44872840466600117460174197353, −9.589745951887708843049111139433, −9.151832852187039350096474211100, −7.902282692434471180932997685423, −6.96275058821148339807861422585, −5.87731792258094176070637748686, −4.93360540861575345215875371222, −4.44512628066207577450837144100, −2.92719178672864026435517936637, −1.39007383625277665227398354058, 0.50530104762353772026587811976, 2.03342971928354531068712483129, 3.69843165028369852947714085145, 4.44399670911608981403041214187, 5.80360218395474453381332410851, 6.50189791410874173929731785908, 7.80694720191220638707701162165, 7.894926212007546373274902154652, 9.218536771077127072542212880715, 9.575336285285912244991607387814

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