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Properties

Label 2-845-65.4-c1-0-5
Degree $2$
Conductor $845$
Sign $-0.310 - 0.950i$
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  + (0.5 − 0.866i)2-s + (−1.73 − i)3-s + (0.500 + 0.866i)4-s + (−1 + 2i)5-s + (−1.73 + 0.999i)6-s + 3·8-s + (0.499 + 0.866i)9-s + (1.23 + 1.86i)10-s + (−1.73 − i)11-s − 2i·12-s + (3.73 − 2.46i)15-s + (0.500 − 0.866i)16-s + 0.999·18-s + (−5.19 + 3i)19-s + (−2.23 + 0.133i)20-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.999 − 0.577i)3-s + (0.250 + 0.433i)4-s + (−0.447 + 0.894i)5-s + (−0.707 + 0.408i)6-s + 1.06·8-s + (0.166 + 0.288i)9-s + (0.389 + 0.590i)10-s + (−0.522 − 0.301i)11-s − 0.577i·12-s + (0.963 − 0.636i)15-s + (0.125 − 0.216i)16-s + 0.235·18-s + (−1.19 + 0.688i)19-s + (−0.499 + 0.0299i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.310 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.310 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.310 - 0.950i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (654, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.310 - 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.280715 + 0.386851i\)
\(L(\frac12)\) \(\approx\) \(0.280715 + 0.386851i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1 - 2i)T \)
13 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (1.73 + i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.73 + i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.19 - 3i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.19 + 3i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6iT - 31T^{2} \)
37 \( 1 + (3 - 5.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.92 - 4i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.19 - 3i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 + 12iT - 53T^{2} \)
59 \( 1 + (-1.73 + i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3 + 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6 - 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.73 - i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + (-6.92 - 4i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3 - 5.19i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.61038438886329925557255815671, −10.24532808416911385588806297173, −8.483695895779263353317278137542, −7.79164783731099335672652765506, −6.79170479506342242482378262667, −6.33584036081625383969250547840, −5.10601737992682492124667528500, −3.90918736324060584143063042439, −3.01856263425949552459097006485, −1.79533690188706566826252265072, 0.22068787063863482785481586546, 2.04254917420239231998828650909, 4.21324359035736445628688098489, 4.60832816706633818897559453664, 5.67612392306827453591885274221, 5.99531000316740756726577885503, 7.32892248586225495767026712274, 8.006188377228106923656463520882, 9.194807195744211191729843091598, 10.09395236488512352480174521322

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