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Properties

Label 2-845-169.116-c1-0-59
Degree $2$
Conductor $845$
Sign $-0.666 - 0.745i$
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  + (−0.373 − 0.421i)2-s + (−0.840 − 2.21i)3-s + (0.202 − 1.67i)4-s + (−0.239 − 0.970i)5-s + (−0.620 + 1.18i)6-s + (2.92 − 2.01i)7-s + (−1.70 + 1.17i)8-s + (−1.95 + 1.73i)9-s + (−0.320 + 0.463i)10-s + (−2.20 + 2.48i)11-s + (−3.87 + 0.954i)12-s + (−1.74 − 3.15i)13-s + (−1.94 − 0.479i)14-s + (−1.95 + 1.34i)15-s + (−2.13 − 0.525i)16-s + (−1.08 − 1.56i)17-s + ⋯
L(s)  = 1  + (−0.264 − 0.298i)2-s + (−0.485 − 1.27i)3-s + (0.101 − 0.835i)4-s + (−0.107 − 0.434i)5-s + (−0.253 + 0.482i)6-s + (1.10 − 0.763i)7-s + (−0.603 + 0.416i)8-s + (−0.652 + 0.578i)9-s + (−0.101 + 0.146i)10-s + (−0.663 + 0.749i)11-s + (−1.11 + 0.275i)12-s + (−0.482 − 0.875i)13-s + (−0.519 − 0.128i)14-s + (−0.503 + 0.347i)15-s + (−0.533 − 0.131i)16-s + (−0.262 − 0.380i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.666 - 0.745i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.666 - 0.745i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.666 - 0.745i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.370193 + 0.827251i\)
\(L(\frac12)\) \(\approx\) \(0.370193 + 0.827251i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.239 + 0.970i)T \)
13 \( 1 + (1.74 + 3.15i)T \)
good2 \( 1 + (0.373 + 0.421i)T + (-0.241 + 1.98i)T^{2} \)
3 \( 1 + (0.840 + 2.21i)T + (-2.24 + 1.98i)T^{2} \)
7 \( 1 + (-2.92 + 2.01i)T + (2.48 - 6.54i)T^{2} \)
11 \( 1 + (2.20 - 2.48i)T + (-1.32 - 10.9i)T^{2} \)
17 \( 1 + (1.08 + 1.56i)T + (-6.02 + 15.8i)T^{2} \)
19 \( 1 - 1.97iT - 19T^{2} \)
23 \( 1 - 2.00T + 23T^{2} \)
29 \( 1 + (-4.81 + 4.26i)T + (3.49 - 28.7i)T^{2} \)
31 \( 1 + (-0.483 + 0.921i)T + (-17.6 - 25.5i)T^{2} \)
37 \( 1 + (0.155 - 0.296i)T + (-21.0 - 30.4i)T^{2} \)
41 \( 1 + (6.23 - 2.36i)T + (30.6 - 27.1i)T^{2} \)
43 \( 1 + (-9.42 + 4.94i)T + (24.4 - 35.3i)T^{2} \)
47 \( 1 + (5.41 - 0.657i)T + (45.6 - 11.2i)T^{2} \)
53 \( 1 + (-4.92 - 7.13i)T + (-18.7 + 49.5i)T^{2} \)
59 \( 1 + (-0.814 - 3.30i)T + (-52.2 + 27.4i)T^{2} \)
61 \( 1 + (-0.602 + 0.872i)T + (-21.6 - 57.0i)T^{2} \)
67 \( 1 + (-7.60 + 0.923i)T + (65.0 - 16.0i)T^{2} \)
71 \( 1 + (12.2 - 4.63i)T + (53.1 - 47.0i)T^{2} \)
73 \( 1 + (-0.898 + 1.01i)T + (-8.79 - 72.4i)T^{2} \)
79 \( 1 + (1.57 + 12.9i)T + (-76.7 + 18.9i)T^{2} \)
83 \( 1 + (-3.75 - 1.42i)T + (62.1 + 55.0i)T^{2} \)
89 \( 1 + 9.21iT - 89T^{2} \)
97 \( 1 + (0.313 - 1.27i)T + (-85.8 - 45.0i)T^{2} \)
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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

−9.961660554223802102419496702407, −8.679631557271390457743525239950, −7.72025329366760241742718986963, −7.29669586369222899019537832313, −6.17627267302930745355021141262, −5.25517126747946202250636291891, −4.54630779540699096027517559122, −2.46630579800005661047598542377, −1.46817312609331820539809747063, −0.52515634645238370994578412573, 2.40549898010432403834627815738, 3.51374410894935119762916371775, 4.59453079305341131701031046472, 5.25189414658650024456227884684, 6.40302543128011040479300591170, 7.40293752513904433693776191547, 8.425687842776092981276459303504, 8.876312823120305117286106282864, 9.874265000372466099033517582651, 10.87243504374674601569852273824

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