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Properties

Label 1-1407-1407.1214-r1-0-0
Degree $1$
Conductor $1407$
Sign $0.391 + 0.920i$
Analytic cond. $151.203$
Root an. cond. $151.203$
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  + (0.723 − 0.690i)2-s + (0.0475 − 0.998i)4-s + (0.995 + 0.0950i)5-s + (−0.654 − 0.755i)8-s + (0.786 − 0.618i)10-s + (0.580 − 0.814i)11-s + (−0.654 + 0.755i)13-s + (−0.995 − 0.0950i)16-s + (−0.888 − 0.458i)17-s + (−0.235 + 0.971i)19-s + (0.142 − 0.989i)20-s + (−0.142 − 0.989i)22-s + (0.786 + 0.618i)23-s + (0.981 + 0.189i)25-s + (0.0475 + 0.998i)26-s + ⋯
L(s)  = 1  + (0.723 − 0.690i)2-s + (0.0475 − 0.998i)4-s + (0.995 + 0.0950i)5-s + (−0.654 − 0.755i)8-s + (0.786 − 0.618i)10-s + (0.580 − 0.814i)11-s + (−0.654 + 0.755i)13-s + (−0.995 − 0.0950i)16-s + (−0.888 − 0.458i)17-s + (−0.235 + 0.971i)19-s + (0.142 − 0.989i)20-s + (−0.142 − 0.989i)22-s + (0.786 + 0.618i)23-s + (0.981 + 0.189i)25-s + (0.0475 + 0.998i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $0.391 + 0.920i$
Analytic conductor: \(151.203\)
Root analytic conductor: \(151.203\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (1214, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1407,\ (1:\ ),\ 0.391 + 0.920i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.216287754 + 0.8043988566i\)
\(L(\frac12)\) \(\approx\) \(1.216287754 + 0.8043988566i\)
\(L(1)\) \(\approx\) \(1.388229604 - 0.4982800554i\)
\(L(1)\) \(\approx\) \(1.388229604 - 0.4982800554i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
67 \( 1 \)
good2 \( 1 + (0.723 - 0.690i)T \)
5 \( 1 + (0.995 + 0.0950i)T \)
11 \( 1 + (0.580 - 0.814i)T \)
13 \( 1 + (-0.654 + 0.755i)T \)
17 \( 1 + (-0.888 - 0.458i)T \)
19 \( 1 + (-0.235 + 0.971i)T \)
23 \( 1 + (0.786 + 0.618i)T \)
29 \( 1 - T \)
31 \( 1 + (-0.327 + 0.945i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.841 + 0.540i)T \)
43 \( 1 + (-0.841 + 0.540i)T \)
47 \( 1 + (-0.786 - 0.618i)T \)
53 \( 1 + (0.0475 - 0.998i)T \)
59 \( 1 + (-0.327 + 0.945i)T \)
61 \( 1 + (0.580 + 0.814i)T \)
71 \( 1 + (-0.841 - 0.540i)T \)
73 \( 1 + (0.995 - 0.0950i)T \)
79 \( 1 + (0.327 + 0.945i)T \)
83 \( 1 + (0.415 + 0.909i)T \)
89 \( 1 + (0.928 + 0.371i)T \)
97 \( 1 + T \)
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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

−20.439809723510501016877571021686, −20.12151285536623662404435964119, −18.80859634871643151241502906889, −17.7701287222812879678173780043, −17.29183084494484634430593311539, −16.89962464301319951011110637191, −15.723472740793569573910235330065, −14.94413948186652998433856234556, −14.58067850021694267223631374968, −13.485017263838323383300590580453, −12.97834230706378539240980534624, −12.39455128363480531036291692711, −11.30462666508003959186724376453, −10.429441419443447825803495697782, −9.32129933760742459960332056666, −8.852241030281202899387949035328, −7.67813513114020316037654432998, −6.85441486956309824328409959533, −6.29069450171545032168043383679, −5.239588877231520843776811575649, −4.750200956940445175455201958155, −3.70265502866546494267556494022, −2.55148569496795000442127854559, −1.89171462066415335519357658503, −0.18159343380315599425109417903, 1.316739988353031004336770827112, 1.91291523644452205168474503595, 2.95365266024529482256100872348, 3.74751997407649454640268665231, 4.87307677761792121606345627146, 5.46348608622470583388516470189, 6.48655681687753290153944000023, 6.91528224874647165314857239862, 8.56011625675782115328360380999, 9.341822930589861213586721951615, 9.92602473180347969504647005278, 10.85594956990680371346258003894, 11.53295102575705529253863360994, 12.27693581290699486057298240636, 13.39589010135414692958232146722, 13.57919951576135601763570573007, 14.58864155470067652353373138897, 14.973166756583630207001471381777, 16.31142165154757121323978874135, 16.86650995589108110464489102177, 17.9013918627280131115291026893, 18.63604296488175500528835573869, 19.33127763671615832575414269305, 20.058354471358807614282965279, 20.99576671841471943179330030059

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