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Properties

Label 1-407-407.318-r0-0-0
Degree $1$
Conductor $407$
Sign $0.970 - 0.242i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $0.970 - 0.242i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (318, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ 0.970 - 0.242i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7523209194 - 0.09270027971i\)
\(L(\frac12)\) \(\approx\) \(0.7523209194 - 0.09270027971i\)
\(L(1)\) \(\approx\) \(0.6922759399 + 0.1194969654i\)
\(L(1)\) \(\approx\) \(0.6922759399 + 0.1194969654i\)

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

−24.36396218708555854133223273919, −23.239822994008487281532647010704, −22.65128130603599089343915853206, −21.58733111937829773091966981060, −21.048554188811910653089297139982, −19.837021371005805636024715629379, −18.86667456460398994423463893428, −18.29374162436042374141213445561, −17.584470204372489888945199867670, −17.00243066194041208450178797546, −15.80939233563424427924944777547, −14.39057789913617706031025508300, −13.49359463632911792371427290376, −12.68803788337329646955683668017, −11.42447300310878659483136065020, −10.99284318047208781360700556253, −10.42260847113715425252803050199, −8.98935584620452936690411541740, −7.92533905844616371939286934203, −7.085311192890667995588926688302, −6.15910261673662327017607544316, −4.74874093989071687325847119908, −3.500645274589137220543992605266, −2.06317001798720952028711589150, −1.37413277102858063023083837212, 0.66594999033948244874547331828, 1.94189794279221845570451413469, 4.28216517158301640623057832991, 4.95905320100843940590938528489, 5.79817392812747961616874116101, 6.67990311827927064666549798656, 8.165006601524897971743411294492, 8.89762519005046841197024990793, 9.5936929960828920127676241590, 10.92825193966159986143646180892, 11.311915798696153568968416914417, 12.79736877312304581672541260662, 13.77130529824191687683996466988, 15.15521730901275199878572636413, 15.484424366511795790421836663626, 16.62357039111140315651876156610, 17.16818468840522002747313359614, 17.90382932079736301976886309944, 18.652003309586380932569095407014, 20.13756801003542495476981599004, 20.78489108095942227692418854666, 21.66745527739123772958636705514, 22.74467308049978118987272129470, 23.68192863989896699908315766098, 24.284615401468505171959085549578

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