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Properties

Label 1-6223-6223.3-r0-0-0
Degree $1$
Conductor $6223$
Sign $0.357 + 0.933i$
Analytic cond. $28.8994$
Root an. cond. $28.8994$
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  + (0.365 + 0.930i)2-s + (0.980 + 0.198i)3-s + (−0.733 + 0.680i)4-s + (−0.733 + 0.680i)5-s + (0.173 + 0.984i)6-s + (−0.900 − 0.433i)8-s + (0.921 + 0.388i)9-s + (−0.900 − 0.433i)10-s + (−0.998 + 0.0498i)11-s + (−0.853 + 0.521i)12-s + (−0.980 − 0.198i)13-s + (−0.853 + 0.521i)15-s + (0.0747 − 0.997i)16-s + (0.797 − 0.603i)17-s + (−0.0249 + 0.999i)18-s − 19-s + ⋯
L(s)  = 1  + (0.365 + 0.930i)2-s + (0.980 + 0.198i)3-s + (−0.733 + 0.680i)4-s + (−0.733 + 0.680i)5-s + (0.173 + 0.984i)6-s + (−0.900 − 0.433i)8-s + (0.921 + 0.388i)9-s + (−0.900 − 0.433i)10-s + (−0.998 + 0.0498i)11-s + (−0.853 + 0.521i)12-s + (−0.980 − 0.198i)13-s + (−0.853 + 0.521i)15-s + (0.0747 − 0.997i)16-s + (0.797 − 0.603i)17-s + (−0.0249 + 0.999i)18-s − 19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.357 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.357 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6223\)    =    \(7^{2} \cdot 127\)
Sign: $0.357 + 0.933i$
Analytic conductor: \(28.8994\)
Root analytic conductor: \(28.8994\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6223} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6223,\ (0:\ ),\ 0.357 + 0.933i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.367874639 + 0.9410386052i\)
\(L(\frac12)\) \(\approx\) \(1.367874639 + 0.9410386052i\)
\(L(1)\) \(\approx\) \(0.9731452900 + 0.7081651633i\)
\(L(1)\) \(\approx\) \(0.9731452900 + 0.7081651633i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
127 \( 1 \)
good2 \( 1 + (0.365 + 0.930i)T \)
3 \( 1 + (0.980 + 0.198i)T \)
5 \( 1 + (-0.733 + 0.680i)T \)
11 \( 1 + (-0.998 + 0.0498i)T \)
13 \( 1 + (-0.980 - 0.198i)T \)
17 \( 1 + (0.797 - 0.603i)T \)
19 \( 1 - T \)
23 \( 1 + (0.661 - 0.749i)T \)
29 \( 1 + (-0.456 + 0.889i)T \)
31 \( 1 + (-0.980 - 0.198i)T \)
37 \( 1 + (-0.969 - 0.246i)T \)
41 \( 1 + (0.998 - 0.0498i)T \)
43 \( 1 + (-0.270 - 0.962i)T \)
47 \( 1 + (-0.826 - 0.563i)T \)
53 \( 1 + (0.853 + 0.521i)T \)
59 \( 1 + (-0.0249 + 0.999i)T \)
61 \( 1 + (0.733 - 0.680i)T \)
67 \( 1 + (0.998 + 0.0498i)T \)
71 \( 1 + (-0.318 - 0.947i)T \)
73 \( 1 + (0.988 - 0.149i)T \)
79 \( 1 + (0.698 + 0.715i)T \)
83 \( 1 + (-0.797 + 0.603i)T \)
89 \( 1 + (-0.222 + 0.974i)T \)
97 \( 1 + (-0.124 + 0.992i)T \)
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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

−17.62886289953293926153775591203, −16.97152365293956140675724376779, −16.00295756507564530456233479598, −15.33840817593595160217006362692, −14.7602633361070121583209821038, −14.323656106216749679149228649, −13.28984300602447799920289023395, −12.85102524897467331432673431242, −12.57181394577108822737104045742, −11.69952670112928989563104663273, −11.03262384429336417548566708826, −10.15026852063874481865229991590, −9.648275084488805347117815432630, −8.92743483373266287285246984216, −8.2497394607634514486763020313, −7.75267551699086814928976152452, −6.96785154742231300906096127880, −5.77936984438505681119494428545, −5.03845385543231455983228109247, −4.42766239470724482300705311926, −3.67752702375290220274600600293, −3.13958520568012521740980989207, −2.251014454989867988180374639189, −1.67869498798900278601952490080, −0.68026504059434045132718985878, 0.442674959447481408669705078815, 2.201219423843766671379191544034, 2.76952885214436828501723454460, 3.47780594938409793378612252357, 4.06714638212085209655672884774, 4.970979244683657646394383064108, 5.37935444251099240145499669096, 6.65631992904720750281080005820, 7.231207100272610455637911418778, 7.59253764689594607872736262287, 8.30697134039431494478913649946, 8.87159678700718916633917619460, 9.726170035179140828504472223543, 10.40732854827396430034859402009, 11.05597967020781239778655185953, 12.359585961480461712605122191613, 12.53289005940735555796166314630, 13.36759067080815361045954075354, 14.16605733077082435308838352788, 14.62838414671772067908267139479, 15.12462972895191797127663415497, 15.547789573626696349841761807973, 16.40383515765520720017649824159, 16.750382259845042210980761491253, 17.8906549242757263165670489074

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