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Properties

Label 1-847-847.194-r0-0-0
Degree $1$
Conductor $847$
Sign $0.538 + 0.842i$
Analytic cond. $3.93345$
Root an. cond. $3.93345$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.999 + 0.0190i)2-s + (−0.913 + 0.406i)3-s + (0.999 + 0.0380i)4-s + (0.935 + 0.353i)5-s + (−0.921 + 0.389i)6-s + (0.998 + 0.0570i)8-s + (0.669 − 0.743i)9-s + (0.928 + 0.371i)10-s + (−0.928 + 0.371i)12-s + (−0.985 + 0.170i)13-s + (−0.998 + 0.0570i)15-s + (0.997 + 0.0760i)16-s + (−0.398 + 0.917i)17-s + (0.683 − 0.730i)18-s + (0.861 + 0.508i)19-s + (0.921 + 0.389i)20-s + ⋯
L(s)  = 1  + (0.999 + 0.0190i)2-s + (−0.913 + 0.406i)3-s + (0.999 + 0.0380i)4-s + (0.935 + 0.353i)5-s + (−0.921 + 0.389i)6-s + (0.998 + 0.0570i)8-s + (0.669 − 0.743i)9-s + (0.928 + 0.371i)10-s + (−0.928 + 0.371i)12-s + (−0.985 + 0.170i)13-s + (−0.998 + 0.0570i)15-s + (0.997 + 0.0760i)16-s + (−0.398 + 0.917i)17-s + (0.683 − 0.730i)18-s + (0.861 + 0.508i)19-s + (0.921 + 0.389i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.538 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.538 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.538 + 0.842i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (194, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ 0.538 + 0.842i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.218395200 + 1.215756042i\)
\(L(\frac12)\) \(\approx\) \(2.218395200 + 1.215756042i\)
\(L(1)\) \(\approx\) \(1.699931378 + 0.4631057787i\)
\(L(1)\) \(\approx\) \(1.699931378 + 0.4631057787i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.999 + 0.0190i)T \)
3 \( 1 + (-0.913 + 0.406i)T \)
5 \( 1 + (0.935 + 0.353i)T \)
13 \( 1 + (-0.985 + 0.170i)T \)
17 \( 1 + (-0.398 + 0.917i)T \)
19 \( 1 + (0.861 + 0.508i)T \)
23 \( 1 + (0.235 + 0.971i)T \)
29 \( 1 + (-0.198 - 0.980i)T \)
31 \( 1 + (0.0665 - 0.997i)T \)
37 \( 1 + (0.272 - 0.962i)T \)
41 \( 1 + (0.0855 + 0.996i)T \)
43 \( 1 + (-0.841 + 0.540i)T \)
47 \( 1 + (0.683 + 0.730i)T \)
53 \( 1 + (0.997 - 0.0760i)T \)
59 \( 1 + (-0.820 - 0.572i)T \)
61 \( 1 + (0.483 - 0.875i)T \)
67 \( 1 + (0.981 - 0.189i)T \)
71 \( 1 + (-0.736 - 0.676i)T \)
73 \( 1 + (-0.179 + 0.983i)T \)
79 \( 1 + (-0.964 + 0.263i)T \)
83 \( 1 + (-0.0285 + 0.999i)T \)
89 \( 1 + (-0.580 + 0.814i)T \)
97 \( 1 + (-0.774 - 0.633i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.10889091237419404690694272941, −21.54047394048750608163695692716, −20.47350129862796565043771392805, −19.907344067392298696329291513357, −18.6659511206436117914301530108, −17.88155406571870545627060482631, −17.02504412288652580249189976432, −16.44344951421138757631353647271, −15.614660855057522140825403766871, −14.49467161054335241740641449634, −13.70109220526277469423317656178, −13.08099774512692054609699573343, −12.25351502321497614729710745651, −11.708861421889281907206278291327, −10.59794577547776692994908111941, −10.02298178984933445905961062708, −8.76030867015293800800108486967, −7.204103916193528571408832027086, −6.904632805877850618221171574108, −5.76646686275038888495328856745, −5.099311894789352462742132732187, −4.59505376316037761154980237863, −2.939673924764372657168179488820, −2.0937438794259812709740966194, −0.97367094044201825013911828666, 1.42855248838820488216807542939, 2.44620933343382477215201862164, 3.62503926997018742916137009777, 4.55338862927718005200991532184, 5.49324346829372759856701256568, 5.99436124273451272182423719612, 6.85729137684383486944335955227, 7.749379651653775999028872392299, 9.5507986776815685224769217293, 9.98785567265238748899111790989, 11.02224071992349289082039511633, 11.599208311608654473633216520750, 12.58174652355078775241919911099, 13.23921278431306859979453280649, 14.20208499836002694944377437073, 14.97596629295578635340732662268, 15.640913277393810309072998579953, 16.768033616389407732888569950750, 17.14290332570622224802958924734, 18.036358243292795047572262850990, 19.103045740972659602130729978516, 20.12439194235669061008998366641, 21.09378956590966278508263246080, 21.62861621422358572998527492260, 22.17199770566580644370603059030

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