↑

Properties

Label 1-2349-2349.355-r0-0-0
Degree $1$
Conductor $2349$
Sign $-0.999 + 0.00720i$
Analytic cond. $10.9087$
Root an. cond. $10.9087$
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  + (0.302 − 0.953i)2-s + (−0.816 − 0.576i)4-s + (0.945 − 0.326i)5-s + (0.908 + 0.418i)7-s + (−0.797 + 0.603i)8-s + (−0.0249 − 0.999i)10-s + (−0.983 − 0.181i)11-s + (−0.441 − 0.897i)13-s + (0.674 − 0.738i)14-s + (0.334 + 0.942i)16-s + (0.173 + 0.984i)17-s + (−0.661 − 0.749i)19-s + (−0.960 − 0.278i)20-s + (−0.470 + 0.882i)22-s + (−0.610 − 0.792i)23-s + ⋯
L(s)  = 1  + (0.302 − 0.953i)2-s + (−0.816 − 0.576i)4-s + (0.945 − 0.326i)5-s + (0.908 + 0.418i)7-s + (−0.797 + 0.603i)8-s + (−0.0249 − 0.999i)10-s + (−0.983 − 0.181i)11-s + (−0.441 − 0.897i)13-s + (0.674 − 0.738i)14-s + (0.334 + 0.942i)16-s + (0.173 + 0.984i)17-s + (−0.661 − 0.749i)19-s + (−0.960 − 0.278i)20-s + (−0.470 + 0.882i)22-s + (−0.610 − 0.792i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00720i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00720i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2349\)    =    \(3^{4} \cdot 29\)
Sign: $-0.999 + 0.00720i$
Analytic conductor: \(10.9087\)
Root analytic conductor: \(10.9087\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2349} (355, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2349,\ (0:\ ),\ -0.999 + 0.00720i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.005534901566 - 1.537255723i\)
\(L(\frac12)\) \(\approx\) \(0.005534901566 - 1.537255723i\)
\(L(1)\) \(\approx\) \(0.9137899654 - 0.7945432260i\)
\(L(1)\) \(\approx\) \(0.9137899654 - 0.7945432260i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.302 - 0.953i)T \)
5 \( 1 + (0.945 - 0.326i)T \)
7 \( 1 + (0.908 + 0.418i)T \)
11 \( 1 + (-0.983 - 0.181i)T \)
13 \( 1 + (-0.441 - 0.897i)T \)
17 \( 1 + (0.173 + 0.984i)T \)
19 \( 1 + (-0.661 - 0.749i)T \)
23 \( 1 + (-0.610 - 0.792i)T \)
31 \( 1 + (0.514 - 0.857i)T \)
37 \( 1 + (0.542 - 0.840i)T \)
41 \( 1 + (-0.686 - 0.727i)T \)
43 \( 1 + (-0.189 - 0.981i)T \)
47 \( 1 + (-0.349 + 0.936i)T \)
53 \( 1 + (0.826 + 0.563i)T \)
59 \( 1 + (0.396 + 0.918i)T \)
61 \( 1 + (-0.816 + 0.576i)T \)
67 \( 1 + (-0.870 - 0.492i)T \)
71 \( 1 + (-0.797 - 0.603i)T \)
73 \( 1 + (-0.411 - 0.911i)T \)
79 \( 1 + (-0.556 - 0.830i)T \)
83 \( 1 + (0.140 + 0.990i)T \)
89 \( 1 + (-0.969 - 0.246i)T \)
97 \( 1 + (-0.528 + 0.848i)T \)
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

−20.09516067810693294281580192015, −18.83512215790149737542231417761, −18.261518184188315224469159002775, −17.775021597943925241184291324106, −17.01376544892740159841425819027, −16.47768449626148823655361576913, −15.624756215213390697695807485968, −14.678010908070258098147966111194, −14.35121303837160802101842887552, −13.583800456727462591881190311323, −13.12816135762149432100731231105, −12.0337001496836571676644988010, −11.339566345558128670691985268215, −10.07980498053376585257572888524, −9.86351856896654319102507703251, −8.71824748584755987391435439200, −8.02160908933659548193534973165, −7.25672989337584680558572555929, −6.62821428454825236371049239164, −5.72021460526899638602823053850, −4.989885019337947792732599125699, −4.48660212754025279904210263237, −3.30590601822073074123045104978, −2.34368116863064491770722944391, −1.37625566519518803255397876714, 0.42593751383914852192719646146, 1.60760033448876837534176554240, 2.34035529770033707918982150085, 2.85387236097789429736268719158, 4.24130204978541132195835566574, 4.83854755488286934971817394263, 5.690183424796812501571789741041, 6.0226554046497982808420335744, 7.59446058803923493382903372325, 8.48050516789945942877555905964, 8.90190209512856143540975783492, 10.05159840324766874459250213294, 10.46549478763126574098915673452, 11.0772916618020512370198871083, 12.19535197264649886237543267670, 12.62737843433335879799110813798, 13.39458857937398641510012316342, 13.93209414052713964772795859153, 15.04546700945948553860350714390, 15.11596416640718017841958823158, 16.559084310680061713561836771972, 17.46091488524905624419102410408, 17.85051262920092761063084622653, 18.487590725069191664149866721157, 19.29650472944260336235630519862

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