↑

Properties

Label 1-107-107.44-r0-0-0
Degree $1$
Conductor $107$
Sign $0.846 - 0.532i$
Analytic cond. $0.496905$
Root an. cond. $0.496905$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.147 + 0.989i)2-s + (0.582 − 0.812i)3-s + (−0.956 + 0.292i)4-s + (−0.630 − 0.776i)5-s + (0.889 + 0.456i)6-s + (−0.0887 − 0.996i)7-s + (−0.430 − 0.902i)8-s + (−0.320 − 0.947i)9-s + (0.674 − 0.737i)10-s + (0.992 − 0.118i)11-s + (−0.320 + 0.947i)12-s + (0.482 + 0.875i)13-s + (0.972 − 0.234i)14-s + (−0.998 + 0.0592i)15-s + (0.829 − 0.558i)16-s + (−0.915 − 0.403i)17-s + ⋯
L(s)  = 1  + (0.147 + 0.989i)2-s + (0.582 − 0.812i)3-s + (−0.956 + 0.292i)4-s + (−0.630 − 0.776i)5-s + (0.889 + 0.456i)6-s + (−0.0887 − 0.996i)7-s + (−0.430 − 0.902i)8-s + (−0.320 − 0.947i)9-s + (0.674 − 0.737i)10-s + (0.992 − 0.118i)11-s + (−0.320 + 0.947i)12-s + (0.482 + 0.875i)13-s + (0.972 − 0.234i)14-s + (−0.998 + 0.0592i)15-s + (0.829 − 0.558i)16-s + (−0.915 − 0.403i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.846 - 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.846 - 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(107\)
Sign: $0.846 - 0.532i$
Analytic conductor: \(0.496905\)
Root analytic conductor: \(0.496905\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{107} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 107,\ (0:\ ),\ 0.846 - 0.532i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.009728914 - 0.2910549355i\)
\(L(\frac12)\) \(\approx\) \(1.009728914 - 0.2910549355i\)
\(L(1)\) \(\approx\) \(1.082320565 - 0.05399753581i\)
\(L(1)\) \(\approx\) \(1.082320565 - 0.05399753581i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad107 \( 1 \)
good2 \( 1 + (0.147 + 0.989i)T \)
3 \( 1 + (0.582 - 0.812i)T \)
5 \( 1 + (-0.630 - 0.776i)T \)
7 \( 1 + (-0.0887 - 0.996i)T \)
11 \( 1 + (0.992 - 0.118i)T \)
13 \( 1 + (0.482 + 0.875i)T \)
17 \( 1 + (-0.915 - 0.403i)T \)
19 \( 1 + (-0.533 - 0.845i)T \)
23 \( 1 + (0.972 + 0.234i)T \)
29 \( 1 + (0.0296 + 0.999i)T \)
31 \( 1 + (0.757 + 0.652i)T \)
37 \( 1 + (-0.794 - 0.606i)T \)
41 \( 1 + (0.937 + 0.348i)T \)
43 \( 1 + (-0.630 + 0.776i)T \)
47 \( 1 + (0.937 - 0.348i)T \)
53 \( 1 + (0.147 - 0.989i)T \)
59 \( 1 + (0.0296 - 0.999i)T \)
61 \( 1 + (-0.0887 + 0.996i)T \)
67 \( 1 + (-0.430 + 0.902i)T \)
71 \( 1 + (0.582 + 0.812i)T \)
73 \( 1 + (0.263 - 0.964i)T \)
79 \( 1 + (0.375 + 0.926i)T \)
83 \( 1 + (-0.984 - 0.176i)T \)
89 \( 1 + (0.889 - 0.456i)T \)
97 \( 1 + (0.674 - 0.737i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−29.99141372676930431049289669640, −28.48207863201029467491868567632, −27.618461849552633790371923754994, −27.020114828509739993983181709929, −25.877934232929685915818844589920, −24.71537049052842938728370541360, −22.84601571410147055887777013660, −22.408967401148878408649639622142, −21.41409937706747199897642854899, −20.37500587046583554469369739603, −19.346947522438838755760577043137, −18.720710403210267733682739429166, −17.26222749050886350687342772857, −15.39356619562026771307685928804, −14.97486276255147358955768023973, −13.74730818177337057368444070680, −12.31943343838958233833023972440, −11.21520468806689497514910602204, −10.32549263662923826190432924079, −9.07280855641485737273968894878, −8.20064550986824071746310059776, −6.02638688310936092848212428043, −4.38797926575118135350382790315, −3.39458304081638199383245956604, −2.30455783053727157429016197201, 1.04970061174845590245360078163, 3.6231551087783006500459513968, 4.63116293875337201217346249888, 6.62176678693456418014224537822, 7.196788971262547127684257535322, 8.61053894504378206464947494407, 9.15752498195150297644586486409, 11.45146949942323235972469471222, 12.79555444095632188816907289133, 13.592391111410568196212674432640, 14.51681442386828789475520481881, 15.80659768078420673290389435587, 16.8647401128240031716512259135, 17.71913459674824357954839028874, 19.20605790440072971410749716683, 19.86029729998652639012467764793, 21.1900049269495589985592382614, 22.85017182140443644374914739143, 23.66855568641007159549048127081, 24.33431303163951456885710072047, 25.22728569486901266109805893436, 26.36394091910708717041751938805, 27.098893204087259682905653173, 28.39938893241202091977143158302, 29.80921781682727573992659116580

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