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Properties

Label 2-845-65.28-c1-0-7
Degree $2$
Conductor $845$
Sign $-0.847 + 0.530i$
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  + (−0.113 + 0.0656i)2-s + (−0.0890 + 0.332i)3-s + (−0.991 + 1.71i)4-s + (−2.08 + 0.813i)5-s + (−0.0116 − 0.0436i)6-s + (−1.39 + 2.40i)7-s − 0.522i·8-s + (2.49 + 1.44i)9-s + (0.183 − 0.229i)10-s + (−1.04 + 3.91i)11-s + (−0.482 − 0.482i)12-s − 0.365i·14-s + (−0.0847 − 0.764i)15-s + (−1.94 − 3.37i)16-s + (2.34 − 0.627i)17-s − 0.378·18-s + ⋯
L(s)  = 1  + (−0.0804 + 0.0464i)2-s + (−0.0513 + 0.191i)3-s + (−0.495 + 0.858i)4-s + (−0.931 + 0.363i)5-s + (−0.00477 − 0.0178i)6-s + (−0.525 + 0.910i)7-s − 0.184i·8-s + (0.831 + 0.480i)9-s + (0.0580 − 0.0724i)10-s + (−0.316 + 1.18i)11-s + (−0.139 − 0.139i)12-s − 0.0976i·14-s + (−0.0218 − 0.197i)15-s + (−0.487 − 0.843i)16-s + (0.567 − 0.152i)17-s − 0.0891·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.847 + 0.530i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.847 + 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.847 + 0.530i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (418, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.847 + 0.530i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.143347 - 0.499454i\)
\(L(\frac12)\) \(\approx\) \(0.143347 - 0.499454i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.08 - 0.813i)T \)
13 \( 1 \)
good2 \( 1 + (0.113 - 0.0656i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (0.0890 - 0.332i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (1.39 - 2.40i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.04 - 3.91i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-2.34 + 0.627i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.83 + 0.491i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (7.70 + 2.06i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-3.96 + 2.28i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.87 - 3.87i)T - 31iT^{2} \)
37 \( 1 + (3.50 + 6.07i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.20 + 1.66i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-1.67 - 6.24i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 - 0.512T + 47T^{2} \)
53 \( 1 + (1.32 + 1.32i)T + 53iT^{2} \)
59 \( 1 + (-0.679 - 2.53i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.641 + 1.11i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.13 + 1.80i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.66 + 6.20i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + 9.93iT - 73T^{2} \)
79 \( 1 - 8.37iT - 79T^{2} \)
83 \( 1 + 3.17T + 83T^{2} \)
89 \( 1 + (-6.01 - 1.61i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-10.1 - 5.88i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.50287249065268210449087569676, −9.831541859579686988260860740361, −9.000665252080827029168173470477, −7.969339384521973686021726227936, −7.51113693735981211105835855208, −6.61113012251247255659261611914, −5.16172626443092777875982525697, −4.32898786399189513349400169895, −3.45821622537722515980828979498, −2.32042357944865347761184731406, 0.29177821090063338201057890855, 1.31197314872825566391898586363, 3.49749773389497222016009400651, 4.08627462609593819639771872080, 5.24594346265161571205562153651, 6.20066916540710044304723409986, 7.15039414038876712199350579301, 8.038321928763751874030440521022, 8.832059385113340941147322292377, 9.949479471445969785366737453318

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