↑

Properties

Label 1-107-107.42-r0-0-0
Degree $1$
Conductor $107$
Sign $-0.521 + 0.852i$
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  + (0.889 + 0.456i)2-s + (−0.205 + 0.978i)3-s + (0.582 + 0.812i)4-s + (−0.956 − 0.292i)5-s + (−0.630 + 0.776i)6-s + (0.0296 + 0.999i)7-s + (0.147 + 0.989i)8-s + (−0.915 − 0.403i)9-s + (−0.717 − 0.696i)10-s + (−0.533 − 0.845i)11-s + (−0.915 + 0.403i)12-s + (0.937 + 0.348i)13-s + (−0.430 + 0.902i)14-s + (0.482 − 0.875i)15-s + (−0.320 + 0.947i)16-s + (0.375 + 0.926i)17-s + ⋯
L(s)  = 1  + (0.889 + 0.456i)2-s + (−0.205 + 0.978i)3-s + (0.582 + 0.812i)4-s + (−0.956 − 0.292i)5-s + (−0.630 + 0.776i)6-s + (0.0296 + 0.999i)7-s + (0.147 + 0.989i)8-s + (−0.915 − 0.403i)9-s + (−0.717 − 0.696i)10-s + (−0.533 − 0.845i)11-s + (−0.915 + 0.403i)12-s + (0.937 + 0.348i)13-s + (−0.430 + 0.902i)14-s + (0.482 − 0.875i)15-s + (−0.320 + 0.947i)16-s + (0.375 + 0.926i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.521 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.521 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(107\)
Sign: $-0.521 + 0.852i$
Analytic conductor: \(0.496905\)
Root analytic conductor: \(0.496905\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{107} (42, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 107,\ (0:\ ),\ -0.521 + 0.852i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6478562516 + 1.156029691i\)
\(L(\frac12)\) \(\approx\) \(0.6478562516 + 1.156029691i\)
\(L(1)\) \(\approx\) \(1.034405498 + 0.8481145476i\)
\(L(1)\) \(\approx\) \(1.034405498 + 0.8481145476i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad107 \( 1 \)
good2 \( 1 + (0.889 + 0.456i)T \)
3 \( 1 + (-0.205 + 0.978i)T \)
5 \( 1 + (-0.956 - 0.292i)T \)
7 \( 1 + (0.0296 + 0.999i)T \)
11 \( 1 + (-0.533 - 0.845i)T \)
13 \( 1 + (0.937 + 0.348i)T \)
17 \( 1 + (0.375 + 0.926i)T \)
19 \( 1 + (0.757 - 0.652i)T \)
23 \( 1 + (-0.430 - 0.902i)T \)
29 \( 1 + (-0.861 + 0.508i)T \)
31 \( 1 + (0.972 + 0.234i)T \)
37 \( 1 + (0.674 - 0.737i)T \)
41 \( 1 + (0.992 + 0.118i)T \)
43 \( 1 + (-0.956 + 0.292i)T \)
47 \( 1 + (0.992 - 0.118i)T \)
53 \( 1 + (0.889 - 0.456i)T \)
59 \( 1 + (-0.861 - 0.508i)T \)
61 \( 1 + (0.0296 - 0.999i)T \)
67 \( 1 + (0.147 - 0.989i)T \)
71 \( 1 + (-0.205 - 0.978i)T \)
73 \( 1 + (-0.0887 + 0.996i)T \)
79 \( 1 + (-0.794 + 0.606i)T \)
83 \( 1 + (-0.998 - 0.0592i)T \)
89 \( 1 + (-0.630 - 0.776i)T \)
97 \( 1 + (-0.717 - 0.696i)T \)
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

−29.64672448680714090023880876157, −28.51586438629146081851008553184, −27.54784711388475745342770218663, −26.02777194859671690302415265142, −24.84284016823477826736145968816, −23.68564251239408026880688888702, −23.12903069199023515214014492299, −22.58925376391338681336197247909, −20.60846479389458075088133017593, −20.1249751597504318187197588377, −18.95839021333022386854905822453, −18.06486227311912289941497926184, −16.41791179755776797942451488225, −15.292423885488909919674871575530, −13.985035658924670474899502940629, −13.231098324911151038456635935886, −12.02771681776578394440055747156, −11.28610442218733889430017463604, −10.11193590187052990933545118365, −7.76379538650038907917980106093, −7.14293633027576097430330056120, −5.69782930891671433023449749864, −4.19307839857974592222053117733, −2.92625388602091300803394662286, −1.14163813874462832766516716853, 2.98640894545436486152263649038, 4.01893375235918430820827986450, 5.22721452484189115175472378931, 6.18703492681111967216733117393, 8.09044168366153714608617943882, 8.87211148848381272502583312743, 10.9150235352987384508822215069, 11.66060553411981396108222403406, 12.77132200719096251540281641001, 14.274168210105830277028908296494, 15.3796042785857888342098142882, 15.94019468964674535147341857787, 16.73244197979953315420633380438, 18.40075885369052464636634719260, 19.903851501682618451575204088480, 21.03954925074066476494028339232, 21.70999849130108652965715231759, 22.75041153858213134441405631866, 23.68731263107582496085266283194, 24.57823651388240852442575476694, 26.008392767510299756771607431536, 26.66025389605466836636971952731, 28.0763042799947675645226559050, 28.65647005324879609672839791753, 30.37997225262554389441621349972

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