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Properties

Label 2-845-65.58-c1-0-66
Degree $2$
Conductor $845$
Sign $-0.905 - 0.424i$
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  + (−0.427 − 0.246i)2-s + (0.908 − 0.243i)3-s + (−0.878 − 1.52i)4-s + (0.284 − 2.21i)5-s + (−0.448 − 0.120i)6-s + (−1.83 − 3.18i)7-s + 1.85i·8-s + (−1.83 + 1.05i)9-s + (−0.669 + 0.878i)10-s + (−0.664 + 0.177i)11-s + (−1.16 − 1.16i)12-s + 1.81i·14-s + (−0.281 − 2.08i)15-s + (−1.29 + 2.24i)16-s + (−0.614 + 2.29i)17-s + 1.04·18-s + ⋯
L(s)  = 1  + (−0.302 − 0.174i)2-s + (0.524 − 0.140i)3-s + (−0.439 − 0.760i)4-s + (0.127 − 0.991i)5-s + (−0.183 − 0.0490i)6-s + (−0.694 − 1.20i)7-s + 0.655i·8-s + (−0.610 + 0.352i)9-s + (−0.211 + 0.277i)10-s + (−0.200 + 0.0536i)11-s + (−0.337 − 0.337i)12-s + 0.485i·14-s + (−0.0726 − 0.538i)15-s + (−0.324 + 0.562i)16-s + (−0.149 + 0.556i)17-s + 0.246·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.905 - 0.424i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.905 - 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.905 - 0.424i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.905 - 0.424i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.141785 + 0.636000i\)
\(L(\frac12)\) \(\approx\) \(0.141785 + 0.636000i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.284 + 2.21i)T \)
13 \( 1 \)
good2 \( 1 + (0.427 + 0.246i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-0.908 + 0.243i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (1.83 + 3.18i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.664 - 0.177i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (0.614 - 2.29i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.41 + 5.29i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.350 + 1.30i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-8.24 - 4.75i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.81 - 4.81i)T - 31iT^{2} \)
37 \( 1 + (-0.917 + 1.58i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.143 + 0.534i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-2.09 - 0.560i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 - 3.80T + 47T^{2} \)
53 \( 1 + (2.47 + 2.47i)T + 53iT^{2} \)
59 \( 1 + (10.0 + 2.69i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (3.09 + 5.36i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.6 + 6.12i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.47 + 1.73i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + 3.37iT - 73T^{2} \)
79 \( 1 + 3.12iT - 79T^{2} \)
83 \( 1 + 2.13T + 83T^{2} \)
89 \( 1 + (-0.874 - 3.26i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-6.12 + 3.53i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.561676756081016220734361532054, −8.949416031443539489812368048587, −8.299694219515305493582904469230, −7.26929642963133938511034673000, −6.18551623522550468875229021520, −5.12145921713171550683598342011, −4.39481483342355958570986817301, −3.05366023045602494354423432102, −1.58526407203430459015873833937, −0.32614270682394806972251029428, 2.58818713210365071542160576364, 3.07016365568598367197415974624, 4.07214072444944193814729251169, 5.69727115956034935076832209417, 6.33122997704478630950608917603, 7.48541549843258920047192565774, 8.171388995562791192373802488120, 9.090568961835906240654095130431, 9.536892025614768893445639533023, 10.39599789848084727321701543312

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