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Properties

Label 1-6304-6304.283-r0-0-0
Degree $1$
Conductor $6304$
Sign $-0.852 + 0.523i$
Analytic cond. $29.2756$
Root an. cond. $29.2756$
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  + (0.476 − 0.879i)3-s + (0.999 − 0.0160i)5-s + (−0.820 − 0.572i)7-s + (−0.545 − 0.838i)9-s + (0.390 − 0.920i)11-s + (0.953 − 0.299i)13-s + (0.462 − 0.886i)15-s + (−0.855 − 0.518i)17-s + (−0.943 + 0.330i)19-s + (−0.893 + 0.448i)21-s + (−0.740 − 0.672i)23-s + (0.999 − 0.0320i)25-s + (−0.996 + 0.0800i)27-s + (0.750 − 0.660i)29-s + (−0.490 + 0.871i)31-s + ⋯
L(s)  = 1  + (0.476 − 0.879i)3-s + (0.999 − 0.0160i)5-s + (−0.820 − 0.572i)7-s + (−0.545 − 0.838i)9-s + (0.390 − 0.920i)11-s + (0.953 − 0.299i)13-s + (0.462 − 0.886i)15-s + (−0.855 − 0.518i)17-s + (−0.943 + 0.330i)19-s + (−0.893 + 0.448i)21-s + (−0.740 − 0.672i)23-s + (0.999 − 0.0320i)25-s + (−0.996 + 0.0800i)27-s + (0.750 − 0.660i)29-s + (−0.490 + 0.871i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.852 + 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.852 + 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6304\)    =    \(2^{5} \cdot 197\)
Sign: $-0.852 + 0.523i$
Analytic conductor: \(29.2756\)
Root analytic conductor: \(29.2756\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6304} (283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6304,\ (0:\ ),\ -0.852 + 0.523i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3549935518 - 1.256432073i\)
\(L(\frac12)\) \(\approx\) \(-0.3549935518 - 1.256432073i\)
\(L(1)\) \(\approx\) \(0.9667729057 - 0.6589013751i\)
\(L(1)\) \(\approx\) \(0.9667729057 - 0.6589013751i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
197 \( 1 \)
good3 \( 1 + (0.476 - 0.879i)T \)
5 \( 1 + (0.999 - 0.0160i)T \)
7 \( 1 + (-0.820 - 0.572i)T \)
11 \( 1 + (0.390 - 0.920i)T \)
13 \( 1 + (0.953 - 0.299i)T \)
17 \( 1 + (-0.855 - 0.518i)T \)
19 \( 1 + (-0.943 + 0.330i)T \)
23 \( 1 + (-0.740 - 0.672i)T \)
29 \( 1 + (0.750 - 0.660i)T \)
31 \( 1 + (-0.490 + 0.871i)T \)
37 \( 1 + (-0.996 - 0.0800i)T \)
41 \( 1 + (0.855 + 0.518i)T \)
43 \( 1 + (-0.920 - 0.390i)T \)
47 \( 1 + (-0.801 - 0.598i)T \)
53 \( 1 + (-0.610 - 0.791i)T \)
59 \( 1 + (0.585 - 0.810i)T \)
61 \( 1 + (0.476 + 0.879i)T \)
67 \( 1 + (-0.143 - 0.989i)T \)
71 \( 1 + (-0.838 + 0.545i)T \)
73 \( 1 + (0.761 - 0.648i)T \)
79 \( 1 + (0.695 + 0.718i)T \)
83 \( 1 + (-0.330 + 0.943i)T \)
89 \( 1 + (-0.871 + 0.490i)T \)
97 \( 1 + (0.949 + 0.315i)T \)
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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

−17.837525119356322311941043614255, −17.38855955890528213264911858863, −16.63208541186150520648566369885, −15.92681408254963289400594529050, −15.480143460387305388284343067084, −14.76029329453583802021378468246, −14.215524135585720611529093394016, −13.35262691030685138245880567630, −13.03945811808247665774315525853, −12.1955685259296191666600137097, −11.245812687222954383723448213081, −10.60418583402166100689430197166, −9.95301722603046086171454832595, −9.43012740802823045868617985551, −8.84694624943000686723897233736, −8.42313997898116804351157537132, −7.208298825825567863988499622890, −6.36835674643765703005565514643, −6.009451492293706382945102628689, −5.11230152584884042582447223452, −4.342896815496225084413646046343, −3.7148911992137755127584152272, −2.863027215770310485086191692975, −2.10740391997103261544231301162, −1.60651491010515913998948516803, 0.27274138311260054234905836703, 1.1363590359285372746553304168, 1.92701919057273447697796398618, 2.68022581207674822136052347326, 3.41759268947875758639701129828, 4.04197149060031308645620289236, 5.23162890462225949144101852916, 6.1226536681637813201513094204, 6.5287503993570675352648902673, 6.84516331104285503298012666507, 8.10307950668225735938662106078, 8.55866084666010343321468701895, 9.147937799927419853402967434142, 9.93959024934466899828354475859, 10.6118693046859325991701648869, 11.27911649375774979875768906186, 12.21892122806413198807371693845, 12.92100060636996000285060011080, 13.330921182023674060396964018185, 13.943832055949521347486538995185, 14.24378004147663639744786023292, 15.21564515634084756927814896449, 16.14088829793478373491955977838, 16.56747169522962298794545713506, 17.40427632121738837267133318524

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