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Properties

Label 1-5077-5077.239-r0-0-0
Degree $1$
Conductor $5077$
Sign $0.0677 - 0.997i$
Analytic cond. $23.5774$
Root an. cond. $23.5774$
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  + (−0.994 + 0.103i)2-s + (0.400 − 0.916i)3-s + (0.978 − 0.206i)4-s + (−0.860 − 0.509i)5-s + (−0.303 + 0.952i)6-s + (−0.282 + 0.959i)7-s + (−0.951 + 0.306i)8-s + (−0.679 − 0.734i)9-s + (0.908 + 0.417i)10-s + (−0.999 + 0.0297i)11-s + (0.202 − 0.979i)12-s + (−0.274 − 0.961i)13-s + (0.180 − 0.983i)14-s + (−0.811 + 0.584i)15-s + (0.914 − 0.404i)16-s + (−0.427 + 0.903i)17-s + ⋯
L(s)  = 1  + (−0.994 + 0.103i)2-s + (0.400 − 0.916i)3-s + (0.978 − 0.206i)4-s + (−0.860 − 0.509i)5-s + (−0.303 + 0.952i)6-s + (−0.282 + 0.959i)7-s + (−0.951 + 0.306i)8-s + (−0.679 − 0.734i)9-s + (0.908 + 0.417i)10-s + (−0.999 + 0.0297i)11-s + (0.202 − 0.979i)12-s + (−0.274 − 0.961i)13-s + (0.180 − 0.983i)14-s + (−0.811 + 0.584i)15-s + (0.914 − 0.404i)16-s + (−0.427 + 0.903i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0677 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0677 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5077\)
Sign: $0.0677 - 0.997i$
Analytic conductor: \(23.5774\)
Root analytic conductor: \(23.5774\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5077} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5077,\ (0:\ ),\ 0.0677 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3244867227 - 0.3031886927i\)
\(L(\frac12)\) \(\approx\) \(0.3244867227 - 0.3031886927i\)
\(L(1)\) \(\approx\) \(0.4975159987 - 0.1371490206i\)
\(L(1)\) \(\approx\) \(0.4975159987 - 0.1371490206i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5077 \( 1 \)
good2 \( 1 + (-0.994 + 0.103i)T \)
3 \( 1 + (0.400 - 0.916i)T \)
5 \( 1 + (-0.860 - 0.509i)T \)
7 \( 1 + (-0.282 + 0.959i)T \)
11 \( 1 + (-0.999 + 0.0297i)T \)
13 \( 1 + (-0.274 - 0.961i)T \)
17 \( 1 + (-0.427 + 0.903i)T \)
19 \( 1 + (-0.0185 - 0.999i)T \)
23 \( 1 + (-0.144 + 0.989i)T \)
29 \( 1 + (0.993 + 0.111i)T \)
31 \( 1 + (-0.506 + 0.862i)T \)
37 \( 1 + (-0.942 + 0.335i)T \)
41 \( 1 + (0.999 - 0.0371i)T \)
43 \( 1 + (-0.875 + 0.483i)T \)
47 \( 1 + (-0.856 - 0.515i)T \)
53 \( 1 + (-0.0779 + 0.996i)T \)
59 \( 1 + (0.852 - 0.522i)T \)
61 \( 1 + (-0.998 - 0.0519i)T \)
67 \( 1 + (-0.386 - 0.922i)T \)
71 \( 1 + (0.303 - 0.952i)T \)
73 \( 1 + (0.944 - 0.328i)T \)
79 \( 1 + (0.246 + 0.969i)T \)
83 \( 1 + (-0.129 + 0.991i)T \)
89 \( 1 + (-0.824 + 0.565i)T \)
97 \( 1 + (-0.645 - 0.763i)T \)
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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

−18.298881599253558926176357167244, −17.50695896073766257803714840890, −16.554551277560609845909040961788, −16.23332746306559497512554517654, −15.87213918448118781454754059729, −14.97273964457516084713933466535, −14.409190873271474563476455918406, −13.72903053723488157779104519520, −12.69109068559909939538158892524, −11.82509319052612428675462103758, −11.18829555610029915592975964006, −10.639749600380390740598969614133, −10.06991323999334245334870408038, −9.591138535952774980024628080907, −8.57619612415567123255577697189, −8.14282716182036779248280779229, −7.34895666430297680133392265471, −6.907937914398296256458150097952, −5.950205193954111285665551247485, −4.75734441591992679567289465488, −4.13662851449150193104652483470, −3.37927695379320497864501828903, −2.71609198759723068852816096384, −1.98347565875094754281937895354, −0.49116642019879831078874802996, 0.30078655385781320907199305075, 1.32706872003659466545985138370, 2.153808953622831866777628138814, 2.95727766675835926515860775574, 3.40050211531713249149086687235, 4.98047570868188227606225694594, 5.54364387119699015465766306746, 6.45200224251245880550727041793, 7.08995814339884348331924752200, 7.984571455774008400823682121911, 8.15064225741357618336778901178, 8.87990053477060450008066039961, 9.44941995990682528083980117519, 10.461206762802057604139671093318, 11.16103344048888311945152774531, 11.87461826709713600406588450962, 12.602603584801042493193022701409, 12.7928163756602990135758742844, 13.79272172821258483717688717627, 14.959284744400786428439082576559, 15.42675857674213711988323991246, 15.65066003086354799137610488333, 16.61129003374737444080411549780, 17.468429013981778816976664256248, 18.01119997536556901559596528027

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