Properties

Label 2-845-845.392-c1-0-15
Degree $2$
Conductor $845$
Sign $0.474 - 0.879i$
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  + (−0.763 + 1.60i)2-s + (−1.71 − 1.58i)3-s + (−0.742 − 0.909i)4-s + (1.90 − 1.16i)5-s + (3.85 − 1.55i)6-s + (1.61 + 0.468i)7-s + (−1.42 + 0.352i)8-s + (0.196 + 2.43i)9-s + (0.423 + 3.96i)10-s + (2.48 + 2.91i)11-s + (−0.165 + 2.73i)12-s + (−2.49 + 2.60i)13-s + (−1.98 + 2.24i)14-s + (−5.11 − 1.01i)15-s + (0.994 − 4.86i)16-s + (−0.609 − 1.10i)17-s + ⋯
L(s)  = 1  + (−0.540 + 1.13i)2-s + (−0.990 − 0.913i)3-s + (−0.371 − 0.454i)4-s + (0.852 − 0.522i)5-s + (1.57 − 0.633i)6-s + (0.611 + 0.177i)7-s + (−0.505 + 0.124i)8-s + (0.0656 + 0.812i)9-s + (0.134 + 1.25i)10-s + (0.748 + 0.879i)11-s + (−0.0477 + 0.789i)12-s + (−0.691 + 0.722i)13-s + (−0.531 + 0.600i)14-s + (−1.32 − 0.261i)15-s + (0.248 − 1.21i)16-s + (−0.147 − 0.268i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.474 - 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.474 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.474 - 0.879i$
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: \((2,\ 845,\ (\ :1/2),\ 0.474 - 0.879i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.861888 + 0.514211i\)
\(L(\frac12)\) \(\approx\) \(0.861888 + 0.514211i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.90 + 1.16i)T \)
13 \( 1 + (2.49 - 2.60i)T \)
good2 \( 1 + (0.763 - 1.60i)T + (-1.26 - 1.54i)T^{2} \)
3 \( 1 + (1.71 + 1.58i)T + (0.241 + 2.99i)T^{2} \)
7 \( 1 + (-1.61 - 0.468i)T + (5.91 + 3.74i)T^{2} \)
11 \( 1 + (-2.48 - 2.91i)T + (-1.76 + 10.8i)T^{2} \)
17 \( 1 + (0.609 + 1.10i)T + (-9.08 + 14.3i)T^{2} \)
19 \( 1 + (-1.18 + 0.318i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.380 + 1.41i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-4.06 - 1.92i)T + (18.3 + 22.4i)T^{2} \)
31 \( 1 + (-0.653 - 0.511i)T + (7.41 + 30.0i)T^{2} \)
37 \( 1 + (-6.04 - 8.04i)T + (-10.2 + 35.5i)T^{2} \)
41 \( 1 + (-6.18 - 5.70i)T + (3.29 + 40.8i)T^{2} \)
43 \( 1 + (0.524 + 3.69i)T + (-41.3 + 11.9i)T^{2} \)
47 \( 1 + (-2.21 + 0.838i)T + (35.1 - 31.1i)T^{2} \)
53 \( 1 + (-4.17 + 6.90i)T + (-24.6 - 46.9i)T^{2} \)
59 \( 1 + (-2.56 - 1.69i)T + (23.1 + 54.2i)T^{2} \)
61 \( 1 + (-3.15 + 3.28i)T + (-2.45 - 60.9i)T^{2} \)
67 \( 1 + (3.83 - 4.70i)T + (-13.4 - 65.6i)T^{2} \)
71 \( 1 + (1.75 - 7.78i)T + (-64.1 - 30.4i)T^{2} \)
73 \( 1 + (-0.241 + 0.349i)T + (-25.8 - 68.2i)T^{2} \)
79 \( 1 + (0.554 - 0.210i)T + (59.1 - 52.3i)T^{2} \)
83 \( 1 + (-4.11 - 7.84i)T + (-47.1 + 68.3i)T^{2} \)
89 \( 1 + (-13.4 - 3.59i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-2.88 + 0.963i)T + (77.5 - 58.2i)T^{2} \)
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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

−10.00705163563158918557267544445, −9.330271131466658596883908940752, −8.513112760756030958378233550177, −7.54448010364914647391646771594, −6.73340821908990617834966497678, −6.36835677191073404599330072270, −5.32444962731394765014953529431, −4.66470494985524353395795906857, −2.30708874641072303219414385476, −1.08766247793433330783148216882, 0.843090124472703496228105817398, 2.28655653345994288741509852120, 3.42744182528549519978854584918, 4.57456481759921420861836201547, 5.75498235970718482956852191921, 6.14734340116108107834847966340, 7.59496474745441707679002618295, 8.897129389548594975207111532874, 9.577892458792064239010679830024, 10.29622552407494422709764821621

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