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Properties

Label 1-140-140.107-r0-0-0
Degree $1$
Conductor $140$
Sign $0.0932 - 0.995i$
Analytic cond. $0.650157$
Root an. cond. $0.650157$
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  + (−0.866 − 0.5i)3-s + (0.5 + 0.866i)9-s + (0.5 − 0.866i)11-s − i·13-s + (−0.866 − 0.5i)17-s + (−0.5 − 0.866i)19-s + (0.866 − 0.5i)23-s − i·27-s − 29-s + (0.5 − 0.866i)31-s + (−0.866 + 0.5i)33-s + (0.866 − 0.5i)37-s + (−0.5 + 0.866i)39-s + 41-s − i·43-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)3-s + (0.5 + 0.866i)9-s + (0.5 − 0.866i)11-s − i·13-s + (−0.866 − 0.5i)17-s + (−0.5 − 0.866i)19-s + (0.866 − 0.5i)23-s − i·27-s − 29-s + (0.5 − 0.866i)31-s + (−0.866 + 0.5i)33-s + (0.866 − 0.5i)37-s + (−0.5 + 0.866i)39-s + 41-s − i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0932 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0932 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.0932 - 0.995i$
Analytic conductor: \(0.650157\)
Root analytic conductor: \(0.650157\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 140,\ (0:\ ),\ 0.0932 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5549004225 - 0.5053756509i\)
\(L(\frac12)\) \(\approx\) \(0.5549004225 - 0.5053756509i\)
\(L(1)\) \(\approx\) \(0.7505466530 - 0.2672889839i\)
\(L(1)\) \(\approx\) \(0.7505466530 - 0.2672889839i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 - iT \)
17 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 - T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 + T \)
43 \( 1 - iT \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (0.866 + 0.5i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 - T \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 - iT \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + iT \)
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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

−28.59801827976346342841216828214, −27.70628462595024739825159539816, −26.82110338550153045136345521664, −25.79795612001864718986928409801, −24.54688862030336100809374695262, −23.47715804101775096815571794824, −22.710273733652280150063051793115, −21.7000889445577804985013195617, −20.87854652812573411597760651376, −19.6224434297293086136157018488, −18.41083253476935770807771757637, −17.31636523571081814516196161250, −16.646451910893270613797310987281, −15.434586976062249851367339712254, −14.56247651106062126943916001755, −12.99797180530393131537783361653, −11.95564201490221801463994209155, −11.030690413711933578862525758766, −9.888611386086824755201390094232, −8.90265082846227201894874618284, −7.114681119853278523698173028877, −6.17992731172889653998489720644, −4.77973203458629693976616266354, −3.85103926647261479735295022907, −1.72537110081802532956545855395, 0.782123120160138973286073838, 2.62984636377830881658444391090, 4.449152061314271455929932376647, 5.71641726462156772126658975684, 6.68883326191434324306562833140, 7.908114383612072025210809625542, 9.25717806607075430207347457143, 10.83745309973678458667081454670, 11.38635838994847137741518175476, 12.78393220716191857398897526276, 13.467357376277537383236421312532, 14.972429877322678592906916120642, 16.16198844541734504474494093235, 17.12401184704793041867237103737, 17.98886371010624541134698843713, 19.00865681234844899091689448956, 19.99403661330658389908611454646, 21.410371043806562429357585580738, 22.37773678417799483419131056726, 23.082395008243954517544157777062, 24.396384917642869804746887868243, 24.76792276868589722751035004552, 26.27139768243418061710008437697, 27.40260224192915497421679948843, 28.14528947355521634900890687653

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