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Properties

Label 1-109-109.63-r0-0-0
Degree $1$
Conductor $109$
Sign $0.0319 + 0.999i$
Analytic cond. $0.506193$
Root an. cond. $0.506193$
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  + 2-s + (−0.5 + 0.866i)3-s + 4-s + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)6-s + (−0.5 + 0.866i)7-s + 8-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.5 − 0.866i)15-s + 16-s + 17-s + ⋯
L(s)  = 1  + 2-s + (−0.5 + 0.866i)3-s + 4-s + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)6-s + (−0.5 + 0.866i)7-s + 8-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)12-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.5 − 0.866i)15-s + 16-s + 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0319 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0319 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(109\)
Sign: $0.0319 + 0.999i$
Analytic conductor: \(0.506193\)
Root analytic conductor: \(0.506193\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{109} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 109,\ (0:\ ),\ 0.0319 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.005075592 + 0.9734398337i\)
\(L(\frac12)\) \(\approx\) \(1.005075592 + 0.9734398337i\)
\(L(1)\) \(\approx\) \(1.245842636 + 0.6323385247i\)
\(L(1)\) \(\approx\) \(1.245842636 + 0.6323385247i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad109 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + T \)
19 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + T \)
43 \( 1 + T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−29.32620084271874461055762152736, −28.719071238222014135726857687628, −27.524792356436159818384331760480, −25.79060436355470238521692013835, −24.8248945317518434738928459913, −23.98706570458723794412482226467, −23.01598178636037736113288290675, −22.68159490233520006903100129403, −20.86113256203408991080974654835, −20.06884508046392457017680222595, −19.19618407756825997168521957561, −17.46440984716658738902100907006, −16.59697705457355205339231539833, −15.584753945356464128103550285426, −14.11531387881027278276067505699, −12.858205131889888922670739548728, −12.59516399119471554208154477730, −11.35169063034365318758592355767, −10.03983966567871178193053638165, −7.73849837478893267448010678264, −7.24473876855692120657892262069, −5.62415033036382388387389115767, −4.702750941550879675406880752030, −3.08632382069988233639910828001, −1.19499042256795969542776613405, 2.82633115859537309942663281937, 3.65281876666265298353829824241, 5.206113582028607963293809691782, 6.09889230947678613621382807045, 7.41730639173958895285964736270, 9.345619512918076613822233114436, 10.71730767604843671096095208991, 11.53070740689162900442729661721, 12.45012964958383243019523204818, 14.13472627519729077727454751523, 14.98120102107647663885240542806, 15.92939536610107803192573542113, 16.61787025223537103218867747608, 18.494700820895180997212643486705, 19.460326416518630822504217501214, 20.97073937657716386490317440289, 21.75642129898479702339562428888, 22.46234917423421784855074811212, 23.307992870206309660298644647, 24.37429838695663431209519768846, 25.8314312681143827623764153928, 26.58252353107002112617144515477, 27.86306433759609971439590887793, 29.01157171215407189285360878524, 29.6440047889690859351162806239

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