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Properties

Label 1-407-407.174-r0-0-0
Degree $1$
Conductor $407$
Sign $0.671 - 0.740i$
Analytic cond. $1.89010$
Root an. cond. $1.89010$
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  + (0.913 − 0.406i)2-s + (−0.978 + 0.207i)3-s + (0.669 − 0.743i)4-s + (0.913 + 0.406i)5-s + (−0.809 + 0.587i)6-s + (0.669 − 0.743i)7-s + (0.309 − 0.951i)8-s + (0.913 − 0.406i)9-s + 10-s + (−0.5 + 0.866i)12-s + (−0.104 + 0.994i)13-s + (0.309 − 0.951i)14-s + (−0.978 − 0.207i)15-s + (−0.104 − 0.994i)16-s + (−0.104 − 0.994i)17-s + (0.669 − 0.743i)18-s + ⋯
L(s)  = 1  + (0.913 − 0.406i)2-s + (−0.978 + 0.207i)3-s + (0.669 − 0.743i)4-s + (0.913 + 0.406i)5-s + (−0.809 + 0.587i)6-s + (0.669 − 0.743i)7-s + (0.309 − 0.951i)8-s + (0.913 − 0.406i)9-s + 10-s + (−0.5 + 0.866i)12-s + (−0.104 + 0.994i)13-s + (0.309 − 0.951i)14-s + (−0.978 − 0.207i)15-s + (−0.104 − 0.994i)16-s + (−0.104 − 0.994i)17-s + (0.669 − 0.743i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $0.671 - 0.740i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (174, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ 0.671 - 0.740i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.944558609 - 0.8618417063i\)
\(L(\frac12)\) \(\approx\) \(1.944558609 - 0.8618417063i\)
\(L(1)\) \(\approx\) \(1.592849708 - 0.4353514868i\)
\(L(1)\) \(\approx\) \(1.592849708 - 0.4353514868i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.913 - 0.406i)T \)
3 \( 1 + (-0.978 + 0.207i)T \)
5 \( 1 + (0.913 + 0.406i)T \)
7 \( 1 + (0.669 - 0.743i)T \)
13 \( 1 + (-0.104 + 0.994i)T \)
17 \( 1 + (-0.104 - 0.994i)T \)
19 \( 1 + (-0.978 + 0.207i)T \)
23 \( 1 + T \)
29 \( 1 + (0.309 + 0.951i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
41 \( 1 + (-0.978 + 0.207i)T \)
43 \( 1 + T \)
47 \( 1 + (0.309 - 0.951i)T \)
53 \( 1 + (0.913 - 0.406i)T \)
59 \( 1 + (-0.978 - 0.207i)T \)
61 \( 1 + (0.913 + 0.406i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.913 + 0.406i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (-0.104 + 0.994i)T \)
83 \( 1 + (-0.104 - 0.994i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.809 - 0.587i)T \)
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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

−24.34123444484849666700608373751, −23.64184308094419538631378595936, −22.698971635852027160508955704512, −21.81744483983290982107402169939, −21.393550891229974861895236741145, −20.56044707402696471047623502869, −19.146576716291848151531211833313, −17.838671481107789302606419726046, −17.39598553140952118857462307581, −16.68202535607869903435343424299, −15.49096555740274969279623563107, −14.87388376060759043584079153587, −13.64377069637539172565343483174, −12.73151282313374298329857786645, −12.37033905708646253044763540165, −11.11564440207308851269031263855, −10.43649818880314380727863010400, −8.85548104863755318473803739083, −7.87003812660499044264966952038, −6.60592500637840238733341606617, −5.78792512222950959448028185229, −5.22812394401159734004961503633, −4.30936750728020424134500039289, −2.57314532828680849030615840996, −1.546867328642699055628365873266, 1.219854481650663314806895081997, 2.257704618760715196023373352252, 3.79636785450408889612208034007, 4.77871462839301702794272410152, 5.46835840453625932446046092892, 6.6874930003532617378620752688, 7.093998414947607200525273117598, 9.2222487816833674767981405082, 10.21605200910072780819577675896, 10.9333183953197796295992721267, 11.53669410802186184183111636611, 12.67029914179088280269790221583, 13.56522379504142783049419665107, 14.35215143276105486858806065989, 15.165428224509928802863507096001, 16.4972940901334752440290132623, 17.012597439689727645423407960432, 18.15909219867109036014577678896, 18.87591846761681682234350184109, 20.248427659257199759469431727591, 21.160284705457224624238236013865, 21.54042464238180547762164741000, 22.47417145710089002962202778905, 23.23175192423316124894212028095, 23.91626428690569662991522894601

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