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Properties

Label 1-6304-6304.381-r1-0-0
Degree $1$
Conductor $6304$
Sign $0.423 - 0.905i$
Analytic cond. $677.458$
Root an. cond. $677.458$
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  + (−0.238 − 0.971i)3-s + (0.143 + 0.989i)5-s + (−0.695 + 0.718i)7-s + (−0.886 + 0.462i)9-s + (0.893 + 0.448i)11-s + (−0.390 + 0.920i)13-s + (0.926 − 0.375i)15-s + (−0.191 + 0.981i)17-s + (−0.111 + 0.993i)19-s + (0.863 + 0.504i)21-s + (−0.938 − 0.345i)23-s + (−0.958 + 0.284i)25-s + (0.660 + 0.750i)27-s + (−0.206 − 0.978i)29-s + (−0.995 + 0.0960i)31-s + ⋯
L(s)  = 1  + (−0.238 − 0.971i)3-s + (0.143 + 0.989i)5-s + (−0.695 + 0.718i)7-s + (−0.886 + 0.462i)9-s + (0.893 + 0.448i)11-s + (−0.390 + 0.920i)13-s + (0.926 − 0.375i)15-s + (−0.191 + 0.981i)17-s + (−0.111 + 0.993i)19-s + (0.863 + 0.504i)21-s + (−0.938 − 0.345i)23-s + (−0.958 + 0.284i)25-s + (0.660 + 0.750i)27-s + (−0.206 − 0.978i)29-s + (−0.995 + 0.0960i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.423 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.423 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6304\)    =    \(2^{5} \cdot 197\)
Sign: $0.423 - 0.905i$
Analytic conductor: \(677.458\)
Root analytic conductor: \(677.458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6304} (381, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6304,\ (1:\ ),\ 0.423 - 0.905i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2547501262 - 0.1621450524i\)
\(L(\frac12)\) \(\approx\) \(0.2547501262 - 0.1621450524i\)
\(L(1)\) \(\approx\) \(0.7596355426 + 0.1236051921i\)
\(L(1)\) \(\approx\) \(0.7596355426 + 0.1236051921i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
197 \( 1 \)
good3 \( 1 + (-0.238 - 0.971i)T \)
5 \( 1 + (0.143 + 0.989i)T \)
7 \( 1 + (-0.695 + 0.718i)T \)
11 \( 1 + (0.893 + 0.448i)T \)
13 \( 1 + (-0.390 + 0.920i)T \)
17 \( 1 + (-0.191 + 0.981i)T \)
19 \( 1 + (-0.111 + 0.993i)T \)
23 \( 1 + (-0.938 - 0.345i)T \)
29 \( 1 + (-0.206 - 0.978i)T \)
31 \( 1 + (-0.995 + 0.0960i)T \)
37 \( 1 + (0.660 - 0.750i)T \)
41 \( 1 + (-0.191 + 0.981i)T \)
43 \( 1 + (0.448 - 0.893i)T \)
47 \( 1 + (0.871 - 0.490i)T \)
53 \( 1 + (0.932 + 0.360i)T \)
59 \( 1 + (-0.791 + 0.610i)T \)
61 \( 1 + (-0.238 + 0.971i)T \)
67 \( 1 + (0.269 - 0.963i)T \)
71 \( 1 + (-0.462 - 0.886i)T \)
73 \( 1 + (-0.997 + 0.0640i)T \)
79 \( 1 + (-0.598 - 0.801i)T \)
83 \( 1 + (0.993 - 0.111i)T \)
89 \( 1 + (-0.0960 + 0.995i)T \)
97 \( 1 + (-0.967 + 0.253i)T \)
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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

−17.44131501776843961210210253281, −16.80121361503699428403015258605, −16.28186657242830978060296831706, −15.854129091844118887651721110083, −15.14836528263202581254691077454, −14.23171729670061027644135710745, −13.76190683607450287488835585430, −12.956023955606257706649438815027, −12.37816424771548539796519008370, −11.55200294088124501919748613778, −11.01679317862981886842851757689, −10.185323438205555156278512796715, −9.57599004303667433994913229938, −9.15450748766717446446030963108, −8.51230038232170452172192233536, −7.55648845687224947575225228126, −6.79910187707919371454108963888, −5.90328492357569853402327358818, −5.392423469330558738416291286816, −4.59976820960908462901173164426, −4.0424872238650539822171172708, −3.32577011945195306159789485403, −2.58073497519837321503181312351, −1.24240172996922024965573177171, −0.50385218369394088798031392360, 0.069687458055317365133631256659, 1.48945031610192452791511838998, 2.09956709444247936453747154254, 2.55660678132615266712030766830, 3.685390449740306854277713559741, 4.17324430355477505613242885869, 5.55426548328048577031016167214, 6.19990127023381799775462761772, 6.37480584502280530396525370211, 7.26398409972110449539078479707, 7.75535674141228852799452430593, 8.7636333371907413167719943241, 9.32726704306421708774368122228, 10.14935704541705335345394089686, 10.77164503675785143469658019820, 11.752780926914638762565989357233, 12.01886554002259362689468930213, 12.623911498713428850581711504957, 13.47574399778362783669519328010, 14.05773011368569943126949473815, 14.81751340478897587927459838976, 15.037815440945462939196021200844, 16.25597401358774474056000117203, 16.76875044147519407662166141717, 17.40872746757696864675704650181

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