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Properties

Label 2-845-65.28-c1-0-43
Degree $2$
Conductor $845$
Sign $0.987 - 0.156i$
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  + (1.37 − 0.792i)2-s + (−0.0510 + 0.190i)3-s + (0.255 − 0.442i)4-s + (2.23 + 0.0672i)5-s + (0.0809 + 0.302i)6-s + (−0.274 + 0.474i)7-s + 2.35i·8-s + (2.56 + 1.48i)9-s + (3.12 − 1.67i)10-s + (0.0396 − 0.147i)11-s + (0.0713 + 0.0713i)12-s + 0.868i·14-s + (−0.126 + 0.422i)15-s + (2.38 + 4.12i)16-s + (−3.03 + 0.813i)17-s + 4.69·18-s + ⋯
L(s)  = 1  + (0.970 − 0.560i)2-s + (−0.0294 + 0.110i)3-s + (0.127 − 0.221i)4-s + (0.999 + 0.0300i)5-s + (0.0330 + 0.123i)6-s + (−0.103 + 0.179i)7-s + 0.834i·8-s + (0.854 + 0.493i)9-s + (0.986 − 0.530i)10-s + (0.0119 − 0.0446i)11-s + (0.0205 + 0.0205i)12-s + 0.232i·14-s + (−0.0327 + 0.109i)15-s + (0.595 + 1.03i)16-s + (−0.736 + 0.197i)17-s + 1.10·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.156i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.987 - 0.156i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (418, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.987 - 0.156i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.94897 + 0.231545i\)
\(L(\frac12)\) \(\approx\) \(2.94897 + 0.231545i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.23 - 0.0672i)T \)
13 \( 1 \)
good2 \( 1 + (-1.37 + 0.792i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (0.0510 - 0.190i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (0.274 - 0.474i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.0396 + 0.147i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (3.03 - 0.813i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (4.40 - 1.18i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.41 - 0.916i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (2.02 - 1.17i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.60 + 6.60i)T - 31iT^{2} \)
37 \( 1 + (3.40 + 5.89i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.45 - 0.926i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (1.84 + 6.86i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 - 9.13T + 47T^{2} \)
53 \( 1 + (3.70 + 3.70i)T + 53iT^{2} \)
59 \( 1 + (0.985 + 3.67i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (3.92 - 6.79i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.23 + 2.44i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.04 + 15.1i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + 3.91iT - 73T^{2} \)
79 \( 1 + 11.1iT - 79T^{2} \)
83 \( 1 + 13.4T + 83T^{2} \)
89 \( 1 + (-8.78 - 2.35i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-6.55 - 3.78i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.61506731167254663662601980635, −9.444178939021150104166061659990, −8.714875971083100195737748393279, −7.59582039962938718315204868751, −6.45587313036957912057017153516, −5.62234427335603430456528500327, −4.71294917191507755181049573882, −3.95849822837942982539647982324, −2.61122444756349376472768001558, −1.83253976670902487318384438658, 1.22455510612709162799641902281, 2.77498609390830597851609265557, 4.15464942146242399447594102623, 4.82062764923303978087094858902, 5.82346078002596773261416076253, 6.73623527416839077933245510500, 6.94872867994310899147990966016, 8.528491271192588476546170971656, 9.437406426790504805879771125767, 10.09997800301411522778980792414

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