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Properties

Label 1-407-407.295-r0-0-0
Degree $1$
Conductor $407$
Sign $0.935 + 0.352i$
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  + (0.809 + 0.587i)2-s + (0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + (0.809 − 0.587i)5-s + (0.809 − 0.587i)6-s + (0.309 + 0.951i)7-s + (−0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s + 10-s + 12-s + (0.809 + 0.587i)13-s + (−0.309 + 0.951i)14-s + (−0.309 − 0.951i)15-s + (−0.809 + 0.587i)16-s + (0.809 − 0.587i)17-s + (−0.309 − 0.951i)18-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + (0.809 − 0.587i)5-s + (0.809 − 0.587i)6-s + (0.309 + 0.951i)7-s + (−0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s + 10-s + 12-s + (0.809 + 0.587i)13-s + (−0.309 + 0.951i)14-s + (−0.309 − 0.951i)15-s + (−0.809 + 0.587i)16-s + (0.809 − 0.587i)17-s + (−0.309 − 0.951i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $0.935 + 0.352i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ 0.935 + 0.352i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.614656288 + 0.4756782211i\)
\(L(\frac12)\) \(\approx\) \(2.614656288 + 0.4756782211i\)
\(L(1)\) \(\approx\) \(1.988453296 + 0.2561588319i\)
\(L(1)\) \(\approx\) \(1.988453296 + 0.2561588319i\)

Euler product

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

−24.000885421352830441297998254890, −23.15902446991322372486045069742, −22.354403578236410001799735163237, −21.625842335610050230063553428005, −20.92549722555941949420159255312, −20.26292306758748671502032619841, −19.42995174636914134698327113247, −18.24449154623414039300160595045, −17.23322839404226756767013991648, −16.1833954687463437742015093760, −15.141672328696912751675402168123, −14.4557658980115841424086031774, −13.72516172692084818710296557598, −13.043660366230038027215314132987, −11.49483573280276648673229949844, −10.63831427177565944250765098061, −10.2662751453975873085324584452, −9.29681318850796030373681488678, −7.898320057471328362359125009198, −6.4630374712844844206528091015, −5.59342884013907000004063401399, −4.53629764853568183358075032502, −3.5923547321931553037523685841, −2.7627251627904922249722160687, −1.430900723774677475447458029971, 1.6463102447829453772953146919, 2.45408769731582527370744640978, 3.781362138253804039108015264514, 5.245995900569836106806443511821, 5.92617272760299569917962798956, 6.67862655254653579346816259204, 8.096672397487344233333643774664, 8.512874528700749801713800726996, 9.671438192433471210022758878645, 11.51560298355656316573510609875, 12.19621691044608709694698265107, 12.88771681650099634575911093132, 13.98114128920057930108807957742, 14.211576650158646898689508379407, 15.49152470263244354436838284381, 16.452095940981031110423004725677, 17.35462023440986631636923176777, 18.199195113328497212047112217866, 18.93377373068790877384387851140, 20.43374128122864945434637457196, 20.962050389849157562073942771021, 21.74844512878801843545375183369, 22.872815881367511349971670603762, 23.66486699267865658531979490476, 24.49553365489097167596213976968

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