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Properties

Label 1-6223-6223.1065-r0-0-0
Degree $1$
Conductor $6223$
Sign $0.445 + 0.895i$
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  + (0.623 − 0.781i)2-s + (0.766 + 0.642i)3-s + (−0.222 − 0.974i)4-s + (0.826 + 0.563i)5-s + (0.980 − 0.198i)6-s + (−0.900 − 0.433i)8-s + (0.173 + 0.984i)9-s + (0.955 − 0.294i)10-s + (−0.939 − 0.342i)11-s + (0.456 − 0.889i)12-s + (0.995 + 0.0995i)13-s + (0.270 + 0.962i)15-s + (−0.900 + 0.433i)16-s + (−0.998 − 0.0498i)17-s + (0.878 + 0.478i)18-s + (−0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (0.623 − 0.781i)2-s + (0.766 + 0.642i)3-s + (−0.222 − 0.974i)4-s + (0.826 + 0.563i)5-s + (0.980 − 0.198i)6-s + (−0.900 − 0.433i)8-s + (0.173 + 0.984i)9-s + (0.955 − 0.294i)10-s + (−0.939 − 0.342i)11-s + (0.456 − 0.889i)12-s + (0.995 + 0.0995i)13-s + (0.270 + 0.962i)15-s + (−0.900 + 0.433i)16-s + (−0.998 − 0.0498i)17-s + (0.878 + 0.478i)18-s + (−0.5 + 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.445 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.445 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6223\)    =    \(7^{2} \cdot 127\)
Sign: $0.445 + 0.895i$
Analytic conductor: \(28.8994\)
Root analytic conductor: \(28.8994\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6223} (1065, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6223,\ (0:\ ),\ 0.445 + 0.895i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.413791827 + 1.494459273i\)
\(L(\frac12)\) \(\approx\) \(2.413791827 + 1.494459273i\)
\(L(1)\) \(\approx\) \(1.797375197 - 0.05510474147i\)
\(L(1)\) \(\approx\) \(1.797375197 - 0.05510474147i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
127 \( 1 \)
good2 \( 1 + (0.623 - 0.781i)T \)
3 \( 1 + (0.766 + 0.642i)T \)
5 \( 1 + (0.826 + 0.563i)T \)
11 \( 1 + (-0.939 - 0.342i)T \)
13 \( 1 + (0.995 + 0.0995i)T \)
17 \( 1 + (-0.998 - 0.0498i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.124 - 0.992i)T \)
29 \( 1 + (-0.853 + 0.521i)T \)
31 \( 1 + (0.980 - 0.198i)T \)
37 \( 1 + (0.980 + 0.198i)T \)
41 \( 1 + (-0.411 + 0.911i)T \)
43 \( 1 + (-0.411 + 0.911i)T \)
47 \( 1 + (0.955 - 0.294i)T \)
53 \( 1 + (0.456 + 0.889i)T \)
59 \( 1 + (0.921 - 0.388i)T \)
61 \( 1 + (-0.988 - 0.149i)T \)
67 \( 1 + (-0.998 + 0.0498i)T \)
71 \( 1 + (0.921 + 0.388i)T \)
73 \( 1 + (-0.988 + 0.149i)T \)
79 \( 1 + (-0.969 - 0.246i)T \)
83 \( 1 + (-0.939 - 0.342i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.921 + 0.388i)T \)
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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

−17.53458085560852756563628452643, −16.99394821379614258060849583752, −15.969025469316526170504267677208, −15.44538128760735042615419492416, −15.0671090633373510129410306557, −14.01896447634421439837813693624, −13.48069606220350019050286440274, −13.26563478385583955458333866267, −12.73308730905511031367502330593, −11.90846993466588144091669120990, −11.09001331452577000659535674761, −10.06175773672272171465642676509, −9.242656520692967904483116225171, −8.674839872896615000999528941326, −8.25920200593993445203604565752, −7.34862631130014474874417273197, −6.852842575465089201705130589207, −5.97219598786677657854344483607, −5.59794220869170770384982448246, −4.576060640294411054577646679875, −4.02598378518642501171074208910, −3.01363124084948113103835319107, −2.36115028902245923155030762642, −1.71003039952532996174715492794, −0.43860817553014842032296894904, 1.25192769556405951975543266314, 2.076334496656346832609794236684, 2.67259691240150132771870425507, 3.179371810729169437145979138137, 4.08972741809643902832982884397, 4.60168289625326069580829442171, 5.54011975349846689802967206344, 6.08856632251778648678567063278, 6.80029096305469981869968401905, 7.991146804044594193994135146394, 8.64216306229956341323408394645, 9.30517088037821012511258649418, 10.02244018784715063358879966266, 10.61519650862604073870652201616, 10.88544841592007902212038129820, 11.7063482312722703442662956929, 12.950798008405245316542258004463, 13.21350120084478724850482728657, 13.74137972456118841564079450263, 14.47661606373099404614580962417, 14.91486219428550185197812972370, 15.600353201239659596684699403764, 16.250151642526951567663558615011, 17.05794232257420848858114351796, 18.28376560928387842870618810390

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