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Properties

Label 2-845-169.114-c1-0-59
Degree $2$
Conductor $845$
Sign $-0.623 - 0.781i$
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  + (−0.635 − 0.518i)2-s + (0.789 + 0.0637i)3-s + (−0.265 − 1.29i)4-s + (0.992 − 0.120i)5-s + (−0.468 − 0.450i)6-s + (−1.53 − 2.42i)7-s + (−1.26 + 2.41i)8-s + (−2.34 − 0.380i)9-s + (−0.693 − 0.438i)10-s + (0.518 + 3.19i)11-s + (−0.126 − 1.04i)12-s + (−3.43 + 1.08i)13-s + (−0.283 + 2.33i)14-s + (0.791 − 0.0318i)15-s + (−0.380 + 0.161i)16-s + (−3.97 + 2.51i)17-s + ⋯
L(s)  = 1  + (−0.449 − 0.366i)2-s + (0.455 + 0.0367i)3-s + (−0.132 − 0.649i)4-s + (0.443 − 0.0539i)5-s + (−0.191 − 0.183i)6-s + (−0.578 − 0.915i)7-s + (−0.448 + 0.854i)8-s + (−0.780 − 0.126i)9-s + (−0.219 − 0.138i)10-s + (0.156 + 0.962i)11-s + (−0.0365 − 0.301i)12-s + (−0.953 + 0.301i)13-s + (−0.0757 + 0.623i)14-s + (0.204 − 0.00823i)15-s + (−0.0950 + 0.0404i)16-s + (−0.963 + 0.609i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.623 - 0.781i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.623 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.623 - 0.781i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (621, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.623 - 0.781i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0194282 + 0.0403636i\)
\(L(\frac12)\) \(\approx\) \(0.0194282 + 0.0403636i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.992 + 0.120i)T \)
13 \( 1 + (3.43 - 1.08i)T \)
good2 \( 1 + (0.635 + 0.518i)T + (0.400 + 1.95i)T^{2} \)
3 \( 1 + (-0.789 - 0.0637i)T + (2.96 + 0.481i)T^{2} \)
7 \( 1 + (1.53 + 2.42i)T + (-3.00 + 6.32i)T^{2} \)
11 \( 1 + (-0.518 - 3.19i)T + (-10.4 + 3.48i)T^{2} \)
17 \( 1 + (3.97 - 2.51i)T + (7.28 - 15.3i)T^{2} \)
19 \( 1 + (4.71 + 2.72i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.11 - 5.39i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.96 + 2.40i)T + (-5.80 - 28.4i)T^{2} \)
31 \( 1 + (0.515 - 2.09i)T + (-27.4 - 14.4i)T^{2} \)
37 \( 1 + (-2.64 + 0.766i)T + (31.2 - 19.7i)T^{2} \)
41 \( 1 + (0.205 - 2.54i)T + (-40.4 - 6.57i)T^{2} \)
43 \( 1 + (-1.20 + 4.16i)T + (-36.3 - 22.9i)T^{2} \)
47 \( 1 + (1.59 - 1.80i)T + (-5.66 - 46.6i)T^{2} \)
53 \( 1 + (4.70 + 2.47i)T + (30.1 + 43.6i)T^{2} \)
59 \( 1 + (-2.72 + 6.39i)T + (-40.8 - 42.5i)T^{2} \)
61 \( 1 + (-0.407 + 10.1i)T + (-60.8 - 4.90i)T^{2} \)
67 \( 1 + (13.2 + 2.70i)T + (61.6 + 26.2i)T^{2} \)
71 \( 1 + (11.2 - 5.33i)T + (44.9 - 54.9i)T^{2} \)
73 \( 1 + (-0.227 - 0.0864i)T + (54.6 + 48.4i)T^{2} \)
79 \( 1 + (0.667 + 0.591i)T + (9.52 + 78.4i)T^{2} \)
83 \( 1 + (0.859 - 0.593i)T + (29.4 - 77.6i)T^{2} \)
89 \( 1 + (-0.0738 + 0.0426i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (11.0 - 14.7i)T + (-26.9 - 93.1i)T^{2} \)
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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

−9.491057442530780667636079048520, −9.222828659790198910934300604721, −8.177997354359403583120434333724, −6.98874326744938273448847849753, −6.31465134474532362117788777100, −5.10735028258880067218867694855, −4.18355508994176372124429142071, −2.72676640464930823669931362124, −1.79033004294878751316700898888, −0.02181659164179730562662170094, 2.58331770363433767079853053010, 2.96862949202627963080137191621, 4.43624287206035092892428030561, 5.77597284238913379361284937465, 6.44315193250537072037649235224, 7.44793320736243955603795388776, 8.608189931306019749842530963175, 8.728234674411917292748734662364, 9.525865840247273943040115521673, 10.58107973377185809191643447750

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