↑

Properties

Label 1-371-371.68-r1-0-0
Degree $1$
Conductor $371$
Sign $-0.999 + 0.0235i$
Analytic cond. $39.8694$
Root an. cond. $39.8694$
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  + (0.799 − 0.600i)2-s + (0.0402 − 0.999i)3-s + (0.278 − 0.960i)4-s + (0.996 + 0.0804i)5-s + (−0.568 − 0.822i)6-s + (−0.354 − 0.935i)8-s + (−0.996 − 0.0804i)9-s + (0.845 − 0.534i)10-s + (−0.200 − 0.979i)11-s + (−0.948 − 0.316i)12-s + (0.970 + 0.239i)13-s + (0.120 − 0.992i)15-s + (−0.845 − 0.534i)16-s + (0.632 + 0.774i)17-s + (−0.845 + 0.534i)18-s + (−0.278 − 0.960i)19-s + ⋯
L(s)  = 1  + (0.799 − 0.600i)2-s + (0.0402 − 0.999i)3-s + (0.278 − 0.960i)4-s + (0.996 + 0.0804i)5-s + (−0.568 − 0.822i)6-s + (−0.354 − 0.935i)8-s + (−0.996 − 0.0804i)9-s + (0.845 − 0.534i)10-s + (−0.200 − 0.979i)11-s + (−0.948 − 0.316i)12-s + (0.970 + 0.239i)13-s + (0.120 − 0.992i)15-s + (−0.845 − 0.534i)16-s + (0.632 + 0.774i)17-s + (−0.845 + 0.534i)18-s + (−0.278 − 0.960i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 371 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 371 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0235i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(371\)    =    \(7 \cdot 53\)
Sign: $-0.999 + 0.0235i$
Analytic conductor: \(39.8694\)
Root analytic conductor: \(39.8694\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{371} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 371,\ (1:\ ),\ -0.999 + 0.0235i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04071430031 - 3.463028123i\)
\(L(\frac12)\) \(\approx\) \(-0.04071430031 - 3.463028123i\)
\(L(1)\) \(\approx\) \(1.187877581 - 1.473740722i\)
\(L(1)\) \(\approx\) \(1.187877581 - 1.473740722i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
53 \( 1 \)
good2 \( 1 + (0.799 - 0.600i)T \)
3 \( 1 + (0.0402 - 0.999i)T \)
5 \( 1 + (0.996 + 0.0804i)T \)
11 \( 1 + (-0.200 - 0.979i)T \)
13 \( 1 + (0.970 + 0.239i)T \)
17 \( 1 + (0.632 + 0.774i)T \)
19 \( 1 + (-0.278 - 0.960i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.748 - 0.663i)T \)
31 \( 1 + (-0.948 + 0.316i)T \)
37 \( 1 + (-0.845 - 0.534i)T \)
41 \( 1 + (0.748 - 0.663i)T \)
43 \( 1 + (0.885 + 0.464i)T \)
47 \( 1 + (-0.428 + 0.903i)T \)
59 \( 1 + (0.996 - 0.0804i)T \)
61 \( 1 + (0.632 - 0.774i)T \)
67 \( 1 + (0.278 - 0.960i)T \)
71 \( 1 + (0.885 + 0.464i)T \)
73 \( 1 + (0.632 + 0.774i)T \)
79 \( 1 + (-0.919 + 0.391i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.987 + 0.160i)T \)
97 \( 1 + (-0.568 + 0.822i)T \)
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

−25.20115636592425764508751740542, −23.84340527320603318144585066486, −22.87207982264028270297630359817, −22.375720835132427034434328439043, −21.33145682808406430567710727175, −20.79381569093919241569379917951, −20.20626910998305195841622560647, −18.325859695031876123964708833520, −17.53031963559818030344309779058, −16.60918573032317460976305445784, −15.97229829793436689231977281849, −14.95213292039817340657738072121, −14.28092999163348609143919418845, −13.41883758764866403560973024657, −12.45088056497702204720544668242, −11.30460674272061137687890332436, −10.21195192304910117024175792213, −9.35529329775025651034397653097, −8.30524461529897462444530861184, −7.09755264441432938700214017014, −5.74325384253518705351227039651, −5.383089048911082979450664034056, −4.155323573455665051139906715039, −3.217323480624171267241884295557, −1.92885728277432230505264215354, 0.68776533344838008111083883427, 1.759436891413421048488585374640, 2.66415397391578742025089767211, 3.79653210531345707454673590029, 5.473623559929911983185699903, 6.0197849167481257309601288395, 6.85705459068871348871721903732, 8.38089356969523844925227896742, 9.35695435755263699132976054732, 10.743157067627268451725712644885, 11.22792068323409325683500164235, 12.61718326285290927771281135295, 13.06463777430924548253422515150, 14.018321999120108568119872800384, 14.41724376878440552238822036148, 15.87179969095917286596403351772, 17.01208451087047607478292249424, 18.09898454480305502536347290755, 18.797169344298789785911894225117, 19.51361003200122640353283238459, 20.68139635661971334524287116710, 21.30081833533379983378242254817, 22.22080617791588383154838654358, 23.084879568039206168384260610361, 24.13221615750908293434154211046

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