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Properties

Label 1-6304-6304.245-r1-0-0
Degree $1$
Conductor $6304$
Sign $0.932 - 0.360i$
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  + (−0.206 − 0.978i)3-s + (−0.684 − 0.729i)5-s + (0.938 + 0.345i)7-s + (−0.914 + 0.404i)9-s + (−0.999 + 0.0160i)11-s + (0.984 − 0.175i)13-s + (−0.572 + 0.820i)15-s + (−0.886 − 0.462i)17-s + (−0.111 − 0.993i)19-s + (0.143 − 0.989i)21-s + (0.995 + 0.0960i)23-s + (−0.0640 + 0.997i)25-s + (0.585 + 0.810i)27-s + (0.610 − 0.791i)29-s + (0.855 + 0.518i)31-s + ⋯
L(s)  = 1  + (−0.206 − 0.978i)3-s + (−0.684 − 0.729i)5-s + (0.938 + 0.345i)7-s + (−0.914 + 0.404i)9-s + (−0.999 + 0.0160i)11-s + (0.984 − 0.175i)13-s + (−0.572 + 0.820i)15-s + (−0.886 − 0.462i)17-s + (−0.111 − 0.993i)19-s + (0.143 − 0.989i)21-s + (0.995 + 0.0960i)23-s + (−0.0640 + 0.997i)25-s + (0.585 + 0.810i)27-s + (0.610 − 0.791i)29-s + (0.855 + 0.518i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6304\)    =    \(2^{5} \cdot 197\)
Sign: $0.932 - 0.360i$
Analytic conductor: \(677.458\)
Root analytic conductor: \(677.458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6304} (245, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6304,\ (1:\ ),\ 0.932 - 0.360i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.607218738 - 0.2994676458i\)
\(L(\frac12)\) \(\approx\) \(1.607218738 - 0.2994676458i\)
\(L(1)\) \(\approx\) \(0.8635201478 - 0.3712818102i\)
\(L(1)\) \(\approx\) \(0.8635201478 - 0.3712818102i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
197 \( 1 \)
good3 \( 1 + (-0.206 - 0.978i)T \)
5 \( 1 + (-0.684 - 0.729i)T \)
7 \( 1 + (0.938 + 0.345i)T \)
11 \( 1 + (-0.999 + 0.0160i)T \)
13 \( 1 + (0.984 - 0.175i)T \)
17 \( 1 + (-0.886 - 0.462i)T \)
19 \( 1 + (-0.111 - 0.993i)T \)
23 \( 1 + (0.995 + 0.0960i)T \)
29 \( 1 + (0.610 - 0.791i)T \)
31 \( 1 + (0.855 + 0.518i)T \)
37 \( 1 + (0.585 - 0.810i)T \)
41 \( 1 + (-0.886 - 0.462i)T \)
43 \( 1 + (-0.0160 - 0.999i)T \)
47 \( 1 + (0.284 - 0.958i)T \)
53 \( 1 + (0.863 + 0.504i)T \)
59 \( 1 + (0.448 + 0.893i)T \)
61 \( 1 + (-0.206 + 0.978i)T \)
67 \( 1 + (0.879 - 0.476i)T \)
71 \( 1 + (0.404 + 0.914i)T \)
73 \( 1 + (0.159 + 0.987i)T \)
79 \( 1 + (-0.999 + 0.0320i)T \)
83 \( 1 + (0.993 + 0.111i)T \)
89 \( 1 + (0.518 + 0.855i)T \)
97 \( 1 + (-0.801 + 0.598i)T \)
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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

−17.50215809216135760768132408276, −16.75711183542782393010822357112, −16.08201712636309550561430543360, −15.50809959156956079469150311733, −14.997534303534975736963042998028, −14.460400925507538947886090521419, −13.73948802529352166146709029638, −12.98065658223730875439338848896, −11.996846366542517663038247089910, −11.26178780444590786828101890444, −10.97545218175987534729295685367, −10.42962880296383755848956081423, −9.794555673139709326935021706860, −8.62877081096971883447109408704, −8.287496793997024421442503173853, −7.66796723796845732807473754504, −6.59114976499179496268233862230, −6.13631245929475399455058016048, −5.03972878836098948915163427373, −4.61417905332190590892038019530, −3.87380728514241951872979838507, −3.22459956852393063765902784439, −2.47600800668929078307655751781, −1.36203674391125909693900185601, −0.32857848160690774623182471087, 0.71709506124520673522608209805, 1.036731883945135659325744704302, 2.244583225307965995509493227718, 2.638202825626940095915797289687, 3.81305982216331780481101499259, 4.749039778529527804660437040579, 5.19428398034103413934382371224, 5.85582933945427452618268051940, 6.92607514798366872545318727302, 7.34022090864953469110155195062, 8.2524840606128697400616193922, 8.55367879026478704449975814919, 9.063824082692535468335001925846, 10.457719828001096950272444823477, 11.070387079368258310738916466467, 11.57746236603826857260898293088, 12.11644094494329801508624335101, 12.91606839234619018827012424341, 13.46653732927043995470789319038, 13.81154085313780993162207067831, 15.03094358148702408618455695059, 15.49076322641769974669355512564, 16.0033638273508355297322429784, 16.98749327100493789564528428827, 17.47943194221417410731957777337

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