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Properties

Label 1-1407-1407.1136-r1-0-0
Degree $1$
Conductor $1407$
Sign $-0.979 + 0.203i$
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  + (−0.0475 + 0.998i)2-s + (−0.995 − 0.0950i)4-s + (−0.981 − 0.189i)5-s + (0.142 − 0.989i)8-s + (0.235 − 0.971i)10-s + (0.327 + 0.945i)11-s + (−0.142 − 0.989i)13-s + (0.981 + 0.189i)16-s + (−0.580 − 0.814i)17-s + (−0.888 − 0.458i)19-s + (0.959 + 0.281i)20-s + (−0.959 + 0.281i)22-s + (−0.235 − 0.971i)23-s + (0.928 + 0.371i)25-s + (0.995 − 0.0950i)26-s + ⋯
L(s)  = 1  + (−0.0475 + 0.998i)2-s + (−0.995 − 0.0950i)4-s + (−0.981 − 0.189i)5-s + (0.142 − 0.989i)8-s + (0.235 − 0.971i)10-s + (0.327 + 0.945i)11-s + (−0.142 − 0.989i)13-s + (0.981 + 0.189i)16-s + (−0.580 − 0.814i)17-s + (−0.888 − 0.458i)19-s + (0.959 + 0.281i)20-s + (−0.959 + 0.281i)22-s + (−0.235 − 0.971i)23-s + (0.928 + 0.371i)25-s + (0.995 − 0.0950i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.979 + 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.979 + 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $-0.979 + 0.203i$
Analytic conductor: \(151.203\)
Root analytic conductor: \(151.203\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (1136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1407,\ (1:\ ),\ -0.979 + 0.203i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01001058080 + 0.09759944626i\)
\(L(\frac12)\) \(\approx\) \(0.01001058080 + 0.09759944626i\)
\(L(1)\) \(\approx\) \(0.6158264564 + 0.1787522216i\)
\(L(1)\) \(\approx\) \(0.6158264564 + 0.1787522216i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
67 \( 1 \)
good2 \( 1 + (-0.0475 + 0.998i)T \)
5 \( 1 + (-0.981 - 0.189i)T \)
11 \( 1 + (0.327 + 0.945i)T \)
13 \( 1 + (-0.142 - 0.989i)T \)
17 \( 1 + (-0.580 - 0.814i)T \)
19 \( 1 + (-0.888 - 0.458i)T \)
23 \( 1 + (-0.235 - 0.971i)T \)
29 \( 1 - T \)
31 \( 1 + (-0.786 - 0.618i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.415 + 0.909i)T \)
43 \( 1 + (0.415 - 0.909i)T \)
47 \( 1 + (-0.235 - 0.971i)T \)
53 \( 1 + (0.995 + 0.0950i)T \)
59 \( 1 + (0.786 + 0.618i)T \)
61 \( 1 + (-0.327 + 0.945i)T \)
71 \( 1 + (-0.415 - 0.909i)T \)
73 \( 1 + (0.981 - 0.189i)T \)
79 \( 1 + (-0.786 + 0.618i)T \)
83 \( 1 + (0.654 - 0.755i)T \)
89 \( 1 + (-0.723 - 0.690i)T \)
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

−20.11297674475395259511437114606, −19.26469374386780247595522439044, −19.162672136496857702591538926398, −18.2982222968882257054743231130, −17.2345190835897960680146542568, −16.638297415362720506395061145363, −15.68043016323530443982141894584, −14.70668721450806840417158680135, −14.1293270984464520140025881025, −13.17241950196957260070530843531, −12.44474369423371078262095199666, −11.59263601569398623750477350490, −11.151541108533139008264008550289, −10.44074180183307556620108381262, −9.32261065154855691686589013155, −8.64649118941705661386693965897, −7.98531192857998007821972635939, −6.90667415239857792315233623709, −5.88862921585739883319965216825, −4.75729563672335548519532268076, −3.81845134537305954323355258950, −3.525330674110042485931829991737, −2.21431652600183525582939983846, −1.33230333286378814505955045131, −0.03299845484306130289711793856, 0.60449375278744505902755075545, 2.20955728232905652321464824431, 3.562126566874605089982232208497, 4.3507879156445898801090274217, 4.98831648103894182698328448319, 5.97893945494773378159944664927, 7.19286883795059872932302466516, 7.29992758221441506099822523970, 8.46659670927465795290210086328, 8.96251712089371624980624153319, 9.98251066524861484514962409128, 10.84325345085774738372197027227, 11.903946972314037504219649095692, 12.72659237248505881432950676727, 13.237344953704094204657973784161, 14.437592828636602189005947321532, 15.13986934994292946251715868346, 15.42349858594889175688756616505, 16.4610731976889810794639331378, 16.95805591695706815160977779734, 17.96242731369446619791519722114, 18.40948195525902178067174835862, 19.47610714181554080477987442447, 20.04017983312172355681903768289, 20.81650324897092570126952368599

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