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Properties

Label 1-1380-1380.119-r0-0-0
Degree $1$
Conductor $1380$
Sign $-0.827 + 0.561i$
Analytic cond. $6.40869$
Root an. cond. $6.40869$
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  + (−0.959 + 0.281i)7-s + (−0.654 − 0.755i)11-s + (0.959 + 0.281i)13-s + (−0.142 + 0.989i)17-s + (0.142 + 0.989i)19-s + (0.142 − 0.989i)29-s + (−0.841 − 0.540i)31-s + (−0.415 + 0.909i)37-s + (−0.415 − 0.909i)41-s + (0.841 − 0.540i)43-s − 47-s + (0.841 − 0.540i)49-s + (−0.959 + 0.281i)53-s + (−0.959 − 0.281i)59-s + (0.841 + 0.540i)61-s + ⋯
L(s)  = 1  + (−0.959 + 0.281i)7-s + (−0.654 − 0.755i)11-s + (0.959 + 0.281i)13-s + (−0.142 + 0.989i)17-s + (0.142 + 0.989i)19-s + (0.142 − 0.989i)29-s + (−0.841 − 0.540i)31-s + (−0.415 + 0.909i)37-s + (−0.415 − 0.909i)41-s + (0.841 − 0.540i)43-s − 47-s + (0.841 − 0.540i)49-s + (−0.959 + 0.281i)53-s + (−0.959 − 0.281i)59-s + (0.841 + 0.540i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.827 + 0.561i$
Analytic conductor: \(6.40869\)
Root analytic conductor: \(6.40869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1380,\ (0:\ ),\ -0.827 + 0.561i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1319673892 + 0.4294591301i\)
\(L(\frac12)\) \(\approx\) \(0.1319673892 + 0.4294591301i\)
\(L(1)\) \(\approx\) \(0.7807533736 + 0.09681824592i\)
\(L(1)\) \(\approx\) \(0.7807533736 + 0.09681824592i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good7 \( 1 + (-0.959 + 0.281i)T \)
11 \( 1 + (-0.654 - 0.755i)T \)
13 \( 1 + (0.959 + 0.281i)T \)
17 \( 1 + (-0.142 + 0.989i)T \)
19 \( 1 + (0.142 + 0.989i)T \)
29 \( 1 + (0.142 - 0.989i)T \)
31 \( 1 + (-0.841 - 0.540i)T \)
37 \( 1 + (-0.415 + 0.909i)T \)
41 \( 1 + (-0.415 - 0.909i)T \)
43 \( 1 + (0.841 - 0.540i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.959 + 0.281i)T \)
59 \( 1 + (-0.959 - 0.281i)T \)
61 \( 1 + (0.841 + 0.540i)T \)
67 \( 1 + (-0.654 + 0.755i)T \)
71 \( 1 + (-0.654 + 0.755i)T \)
73 \( 1 + (0.142 + 0.989i)T \)
79 \( 1 + (0.959 + 0.281i)T \)
83 \( 1 + (-0.415 + 0.909i)T \)
89 \( 1 + (-0.841 + 0.540i)T \)
97 \( 1 + (-0.415 - 0.909i)T \)
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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

−20.37416111294619059493670816221, −19.91560473733986211853196582883, −19.053302574013100839677305615131, −18.02861074381638557439347522399, −17.83911852961503168535551245605, −16.48769718928696990691894961112, −16.0193988761097026435134765213, −15.39664083278845189341074735607, −14.383149720930647291936854123653, −13.4622991473078106243397758182, −12.96474633552628778141652350175, −12.21750891720761572315550114823, −11.07651172105988741498925627201, −10.54864239827699965163278713138, −9.51684436185627519165066775670, −9.03854622085469456598060418979, −7.85648711736480311983974968051, −7.07186228533500456681853986145, −6.408666274436387595043384168783, −5.32571139726978664033477138971, −4.5658004119470672053720049475, −3.3876457301431334775275122371, −2.79949247496100917327011094270, −1.52497453834748647125441263724, −0.17354463474725782556281076077, 1.34251859358901591169774098744, 2.48027986242274890342887263469, 3.48100543854144803403759880156, 4.055950574230364013587227117333, 5.53791519394919807183276413456, 6.000253076202909958099872671353, 6.81202125611783985793398960288, 8.010533291159320653368827484699, 8.5672811170512091048920899404, 9.508662938955958340037721173658, 10.32703224090516639222765988554, 11.02465015715696561267494718843, 11.967251375503034381984411536636, 12.82231454396670401856510934530, 13.40462116585294226444510005925, 14.16254542587897078522054187630, 15.25194556949616384303399003223, 15.848979577676832209529150932373, 16.50359582505017186088671097990, 17.25097296441269856531308598828, 18.36730809772285948517951188916, 18.89380770830823136954111182973, 19.402370705722965083517982892863, 20.550448964825483568123788902982, 21.04452677790706435779876918364

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