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Properties

Label 1-407-407.158-r0-0-0
Degree $1$
Conductor $407$
Sign $-0.345 - 0.938i$
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  + (0.669 − 0.743i)2-s + (0.913 − 0.406i)3-s + (−0.104 − 0.994i)4-s + (0.669 + 0.743i)5-s + (0.309 − 0.951i)6-s + (−0.104 − 0.994i)7-s + (−0.809 − 0.587i)8-s + (0.669 − 0.743i)9-s + 10-s + (−0.5 − 0.866i)12-s + (−0.978 − 0.207i)13-s + (−0.809 − 0.587i)14-s + (0.913 + 0.406i)15-s + (−0.978 + 0.207i)16-s + (−0.978 + 0.207i)17-s + (−0.104 − 0.994i)18-s + ⋯
L(s)  = 1  + (0.669 − 0.743i)2-s + (0.913 − 0.406i)3-s + (−0.104 − 0.994i)4-s + (0.669 + 0.743i)5-s + (0.309 − 0.951i)6-s + (−0.104 − 0.994i)7-s + (−0.809 − 0.587i)8-s + (0.669 − 0.743i)9-s + 10-s + (−0.5 − 0.866i)12-s + (−0.978 − 0.207i)13-s + (−0.809 − 0.587i)14-s + (0.913 + 0.406i)15-s + (−0.978 + 0.207i)16-s + (−0.978 + 0.207i)17-s + (−0.104 − 0.994i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.345 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.345 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $-0.345 - 0.938i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ -0.345 - 0.938i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.487506653 - 2.132668442i\)
\(L(\frac12)\) \(\approx\) \(1.487506653 - 2.132668442i\)
\(L(1)\) \(\approx\) \(1.588748598 - 1.181321911i\)
\(L(1)\) \(\approx\) \(1.588748598 - 1.181321911i\)

Euler product

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

−24.53671276711869347022534643565, −24.33589516221509926849335340006, −22.56025570546063988719804075297, −22.05113964779140663033279840938, −21.15445197892475801708942793305, −20.64831889540570203002744481029, −19.53992097751416518047066516558, −18.39080735784456589474615687251, −17.36182864963774049532339036682, −16.45266377999037353401261235115, −15.654281825673976785282857783651, −14.90587198731501644846884380624, −14.08667520778676888870075321564, −13.186354134046228320881527060806, −12.567849650835487493439198547865, −11.42416148853879318930542207167, −9.58912376267602972442272857995, −9.22262762304309187396239684970, −8.25823673503896176099527379151, −7.28683145951480578408008268242, −5.97054221814794761745598527416, −5.068637153935997359806173567019, −4.30687069340977037979254087331, −2.87857973594147747935427630852, −2.12314352620896458717734849770, 1.18882718426305384039298000504, 2.38004309832875287143608498546, 3.108032845623124812638005553437, 4.144209339980594592537874925711, 5.381123524064279413836679936432, 6.84936189814790519171980799705, 7.20158113526815381728996784679, 8.939707837967848595463569952961, 9.80663847188385493799346488600, 10.53373546100836225341023083384, 11.53103424687902650353532241269, 12.93924620865475833367467383175, 13.28988836267256412468595257102, 14.31697884290858219426924564482, 14.66872906666982758643980184536, 15.79147636185529455833184769861, 17.44125392145551410402597592576, 18.10810485374463685338562418762, 19.22847943195061318340497908750, 19.74737303215641330605773012602, 20.56062802408411774006256096919, 21.38335564448893412082460781994, 22.29110495659393039425536842938, 22.99899431416203290128858357491, 24.16925235949475842106594444567

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