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Properties

Label 1-407-407.153-r0-0-0
Degree $1$
Conductor $407$
Sign $0.739 - 0.673i$
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  + (0.642 + 0.766i)2-s + (−0.766 − 0.642i)3-s + (−0.173 + 0.984i)4-s + (−0.342 − 0.939i)5-s − i·6-s + (0.939 − 0.342i)7-s + (−0.866 + 0.5i)8-s + (0.173 + 0.984i)9-s + (0.5 − 0.866i)10-s + (0.766 − 0.642i)12-s + (−0.984 − 0.173i)13-s + (0.866 + 0.5i)14-s + (−0.342 + 0.939i)15-s + (−0.939 − 0.342i)16-s + (0.984 − 0.173i)17-s + (−0.642 + 0.766i)18-s + ⋯
L(s)  = 1  + (0.642 + 0.766i)2-s + (−0.766 − 0.642i)3-s + (−0.173 + 0.984i)4-s + (−0.342 − 0.939i)5-s − i·6-s + (0.939 − 0.342i)7-s + (−0.866 + 0.5i)8-s + (0.173 + 0.984i)9-s + (0.5 − 0.866i)10-s + (0.766 − 0.642i)12-s + (−0.984 − 0.173i)13-s + (0.866 + 0.5i)14-s + (−0.342 + 0.939i)15-s + (−0.939 − 0.342i)16-s + (0.984 − 0.173i)17-s + (−0.642 + 0.766i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $0.739 - 0.673i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ 0.739 - 0.673i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.145706149 - 0.4433792488i\)
\(L(\frac12)\) \(\approx\) \(1.145706149 - 0.4433792488i\)
\(L(1)\) \(\approx\) \(1.093870485 + 0.02019058481i\)
\(L(1)\) \(\approx\) \(1.093870485 + 0.02019058481i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.642 + 0.766i)T \)
3 \( 1 + (-0.766 - 0.642i)T \)
5 \( 1 + (-0.342 - 0.939i)T \)
7 \( 1 + (0.939 - 0.342i)T \)
13 \( 1 + (-0.984 - 0.173i)T \)
17 \( 1 + (0.984 - 0.173i)T \)
19 \( 1 + (0.642 - 0.766i)T \)
23 \( 1 + (-0.866 - 0.5i)T \)
29 \( 1 + (0.866 - 0.5i)T \)
31 \( 1 - iT \)
41 \( 1 + (0.173 - 0.984i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.939 - 0.342i)T \)
59 \( 1 + (0.342 - 0.939i)T \)
61 \( 1 + (0.984 + 0.173i)T \)
67 \( 1 + (0.939 - 0.342i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + T \)
79 \( 1 + (0.342 + 0.939i)T \)
83 \( 1 + (-0.173 - 0.984i)T \)
89 \( 1 + (-0.342 + 0.939i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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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

−23.97942655955303874724939820909, −23.32382002113791895887402735074, −22.59692774965603463265120453328, −21.63622808638339223423681673285, −21.44640207177591525312871097875, −20.25924902174014994131549687905, −19.31236792171328923012046934591, −18.24695715726625797938723415630, −17.79030900270497019308068007721, −16.374875462497208946029328412010, −15.38960744368206896410292948609, −14.54052284794482565837625781052, −14.17488158471620257697097097664, −12.35470730948266334237645456368, −11.8887730545671988585761751266, −11.14172296815639202402772447051, −10.23203315827839613885523159444, −9.646799072309638720822731595149, −8.01750987629001250686104082925, −6.714103124193191783123599155403, −5.63081001647558375773474494612, −4.88133401092634633333355394279, −3.83123839389016324688078180412, −2.87093431661281736020501439587, −1.445034538609808229970428504, 0.68885225717361987772721869455, 2.28708936422806407084493043356, 4.06918518235110134381843738674, 5.005319214647907010520928508841, 5.46959955810646944096165507377, 6.8162204788956688336398628009, 7.78424108967860700422501918374, 8.17397064796876061445733323815, 9.70023695461302904820201544549, 11.24068154606332440525268352016, 12.03799903451780299198861678735, 12.55259889878169260818081685136, 13.64025139903556394683465784309, 14.34417170235817742655973288906, 15.58642588514343744856994277299, 16.38634331216208780305751507883, 17.27081638919454936594323751554, 17.591927622229736865644084594971, 18.81868663161658007917169998059, 20.06379571026310175438656368642, 20.893263958701556308696871614982, 21.88509043518654569746058872381, 22.711966484434180067342255890444, 23.63281394614732249618312111183, 24.208323965008223300177205538680

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