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Properties

Label 1-407-407.256-r0-0-0
Degree $1$
Conductor $407$
Sign $-0.215 + 0.976i$
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  + (0.990 + 0.139i)2-s + (0.438 + 0.898i)3-s + (0.961 + 0.275i)4-s + (−0.374 + 0.927i)5-s + (0.309 + 0.951i)6-s + (−0.241 − 0.970i)7-s + (0.913 + 0.406i)8-s + (−0.615 + 0.788i)9-s + (−0.5 + 0.866i)10-s + (0.173 + 0.984i)12-s + (0.0348 + 0.999i)13-s + (−0.104 − 0.994i)14-s + (−0.997 + 0.0697i)15-s + (0.848 + 0.529i)16-s + (0.0348 − 0.999i)17-s + (−0.719 + 0.694i)18-s + ⋯
L(s)  = 1  + (0.990 + 0.139i)2-s + (0.438 + 0.898i)3-s + (0.961 + 0.275i)4-s + (−0.374 + 0.927i)5-s + (0.309 + 0.951i)6-s + (−0.241 − 0.970i)7-s + (0.913 + 0.406i)8-s + (−0.615 + 0.788i)9-s + (−0.5 + 0.866i)10-s + (0.173 + 0.984i)12-s + (0.0348 + 0.999i)13-s + (−0.104 − 0.994i)14-s + (−0.997 + 0.0697i)15-s + (0.848 + 0.529i)16-s + (0.0348 − 0.999i)17-s + (−0.719 + 0.694i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $-0.215 + 0.976i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ -0.215 + 0.976i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.594540590 + 1.983949867i\)
\(L(\frac12)\) \(\approx\) \(1.594540590 + 1.983949867i\)
\(L(1)\) \(\approx\) \(1.682990023 + 1.015681064i\)
\(L(1)\) \(\approx\) \(1.682990023 + 1.015681064i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.990 + 0.139i)T \)
3 \( 1 + (0.438 + 0.898i)T \)
5 \( 1 + (-0.374 + 0.927i)T \)
7 \( 1 + (-0.241 - 0.970i)T \)
13 \( 1 + (0.0348 + 0.999i)T \)
17 \( 1 + (0.0348 - 0.999i)T \)
19 \( 1 + (0.438 + 0.898i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.104 + 0.994i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
41 \( 1 + (0.559 - 0.829i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.104 - 0.994i)T \)
53 \( 1 + (-0.374 - 0.927i)T \)
59 \( 1 + (-0.997 + 0.0697i)T \)
61 \( 1 + (-0.615 - 0.788i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (0.990 - 0.139i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (0.848 - 0.529i)T \)
83 \( 1 + (0.0348 - 0.999i)T \)
89 \( 1 + (0.766 - 0.642i)T \)
97 \( 1 + (0.669 + 0.743i)T \)
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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

−24.25782442201321049725584809318, −23.28810034523337617205297134991, −22.49476357686584880107164621933, −21.43612721402990118209407600240, −20.58606116279239230351494925160, −19.756793582298601576513979002077, −19.27777014472636099907396702853, −18.07677118728340028999251925380, −17.0251177099694857209697388388, −15.74347140705008072372502796189, −15.280643222265779871468468166159, −14.20780945577453663898749909105, −13.171316732860522048898030666364, −12.56373667453361371934510114721, −12.1041414255231401906996742942, −10.997651340385054987131776955687, −9.49880493189668434043193066670, −8.390506449990135023341305326874, −7.68501028190823256480273316938, −6.32648323347489904577231123222, −5.654743882697908371539845096955, −4.49342142332185333069216116239, −3.2167130231143525007687758359, −2.339785678784860327912982774120, −1.07349122795789432519397734826, 2.10750520278008259151205993167, 3.35917433984102469485027307506, 3.83130628463439415537319441401, 4.820174635028175079361858179972, 6.086780324967776293405731663201, 7.20825604026325349973312771234, 7.81690893989106940064671913113, 9.45215643386927964803075578458, 10.38220092045110159879616342449, 11.18151220512589269103979132039, 11.94159184925448989644777585365, 13.49979336885242131814775601755, 14.12344598309430922827123901096, 14.6219933520449629647398996371, 15.81915516456101606225105607397, 16.21436547347571850176770675787, 17.21928459778173673325755478771, 18.75019487293120927318844096504, 19.70021033327640246216056805285, 20.39960715847834998537031224626, 21.21086644277115046157874963970, 22.11324793042096618742624446027, 22.7596393233339290190407442865, 23.40875253647298359060947625898, 24.43545106086295306245643332510

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