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Properties

Label 2-845-169.90-c1-0-18
Degree $2$
Conductor $845$
Sign $-0.987 - 0.155i$
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  + (−1.27 + 2.43i)2-s + (2.92 + 0.719i)3-s + (−3.14 − 4.56i)4-s + (0.935 − 0.354i)5-s + (−5.47 + 6.18i)6-s + (−3.37 + 0.410i)7-s + (9.66 − 1.17i)8-s + (5.35 + 2.80i)9-s + (−0.331 + 2.72i)10-s + (−1.42 − 2.71i)11-s + (−5.91 − 15.5i)12-s + (1.31 + 3.35i)13-s + (3.31 − 8.74i)14-s + (2.98 − 0.362i)15-s + (−5.55 + 14.6i)16-s + (0.763 + 6.28i)17-s + ⋯
L(s)  = 1  + (−0.902 + 1.71i)2-s + (1.68 + 0.415i)3-s + (−1.57 − 2.28i)4-s + (0.418 − 0.158i)5-s + (−2.23 + 2.52i)6-s + (−1.27 + 0.155i)7-s + (3.41 − 0.414i)8-s + (1.78 + 0.936i)9-s + (−0.104 + 0.862i)10-s + (−0.429 − 0.819i)11-s + (−1.70 − 4.50i)12-s + (0.364 + 0.931i)13-s + (0.886 − 2.33i)14-s + (0.770 − 0.0936i)15-s + (−1.38 + 3.65i)16-s + (0.185 + 1.52i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.155i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 - 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.987 - 0.155i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (766, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.987 - 0.155i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.111231 + 1.41947i\)
\(L(\frac12)\) \(\approx\) \(0.111231 + 1.41947i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.935 + 0.354i)T \)
13 \( 1 + (-1.31 - 3.35i)T \)
good2 \( 1 + (1.27 - 2.43i)T + (-1.13 - 1.64i)T^{2} \)
3 \( 1 + (-2.92 - 0.719i)T + (2.65 + 1.39i)T^{2} \)
7 \( 1 + (3.37 - 0.410i)T + (6.79 - 1.67i)T^{2} \)
11 \( 1 + (1.42 + 2.71i)T + (-6.24 + 9.05i)T^{2} \)
17 \( 1 + (-0.763 - 6.28i)T + (-16.5 + 4.06i)T^{2} \)
19 \( 1 - 7.21iT - 19T^{2} \)
23 \( 1 - 3.92T + 23T^{2} \)
29 \( 1 + (1.27 + 0.671i)T + (16.4 + 23.8i)T^{2} \)
31 \( 1 + (-0.874 + 0.986i)T + (-3.73 - 30.7i)T^{2} \)
37 \( 1 + (-1.58 + 1.78i)T + (-4.45 - 36.7i)T^{2} \)
41 \( 1 + (1.84 - 7.50i)T + (-36.3 - 19.0i)T^{2} \)
43 \( 1 + (-3.49 + 3.09i)T + (5.18 - 42.6i)T^{2} \)
47 \( 1 + (-3.64 - 2.51i)T + (16.6 + 43.9i)T^{2} \)
53 \( 1 + (0.172 + 1.41i)T + (-51.4 + 12.6i)T^{2} \)
59 \( 1 + (6.11 - 2.31i)T + (44.1 - 39.1i)T^{2} \)
61 \( 1 + (0.758 - 6.24i)T + (-59.2 - 14.5i)T^{2} \)
67 \( 1 + (9.20 + 6.35i)T + (23.7 + 62.6i)T^{2} \)
71 \( 1 + (-0.887 + 3.59i)T + (-62.8 - 32.9i)T^{2} \)
73 \( 1 + (-0.299 - 0.571i)T + (-41.4 + 60.0i)T^{2} \)
79 \( 1 + (-6.57 + 9.52i)T + (-28.0 - 73.8i)T^{2} \)
83 \( 1 + (1.83 + 7.44i)T + (-73.4 + 38.5i)T^{2} \)
89 \( 1 - 1.07iT - 89T^{2} \)
97 \( 1 + (8.44 + 3.20i)T + (72.6 + 64.3i)T^{2} \)
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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

−10.05688677229891531231185633930, −9.303519523992649768228915873435, −8.851831949521293631290142957041, −8.181639640217778962472091678238, −7.48157831881738528094716056331, −6.29591740441494526416043764968, −5.87150066698222574858732931763, −4.35252958107676035571883032168, −3.37371361210845260153478335595, −1.65737896065941690294597942486, 0.806731664633124781399171126943, 2.35117022858984666195269913537, 2.86965996521342402439686280710, 3.45450332940228068506257762991, 4.78556322833876392711117638188, 7.06382459310843732772133506486, 7.44670280178037750361740597367, 8.574372177894859953524861474771, 9.245129830371884528665413057161, 9.632955346979721215671303845656

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