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Properties

Label 1-403-403.64-r0-0-0
Degree $1$
Conductor $403$
Sign $-0.800 + 0.599i$
Analytic cond. $1.87152$
Root an. cond. $1.87152$
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  + (−0.309 − 0.951i)2-s + (0.309 − 0.951i)3-s + (−0.809 + 0.587i)4-s − 5-s − 6-s + (0.809 − 0.587i)7-s + (0.809 + 0.587i)8-s + (−0.809 − 0.587i)9-s + (0.309 + 0.951i)10-s + (0.809 − 0.587i)11-s + (0.309 + 0.951i)12-s + (−0.809 − 0.587i)14-s + (−0.309 + 0.951i)15-s + (0.309 − 0.951i)16-s + (−0.809 − 0.587i)17-s + (−0.309 + 0.951i)18-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)2-s + (0.309 − 0.951i)3-s + (−0.809 + 0.587i)4-s − 5-s − 6-s + (0.809 − 0.587i)7-s + (0.809 + 0.587i)8-s + (−0.809 − 0.587i)9-s + (0.309 + 0.951i)10-s + (0.809 − 0.587i)11-s + (0.309 + 0.951i)12-s + (−0.809 − 0.587i)14-s + (−0.309 + 0.951i)15-s + (0.309 − 0.951i)16-s + (−0.809 − 0.587i)17-s + (−0.309 + 0.951i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.800 + 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.800 + 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $-0.800 + 0.599i$
Analytic conductor: \(1.87152\)
Root analytic conductor: \(1.87152\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{403} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 403,\ (0:\ ),\ -0.800 + 0.599i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2346935003 - 0.7045075130i\)
\(L(\frac12)\) \(\approx\) \(-0.2346935003 - 0.7045075130i\)
\(L(1)\) \(\approx\) \(0.4342368384 - 0.6374793979i\)
\(L(1)\) \(\approx\) \(0.4342368384 - 0.6374793979i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.309 - 0.951i)T \)
3 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 - T \)
7 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 + (0.809 - 0.587i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
23 \( 1 + (-0.809 - 0.587i)T \)
29 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.309 - 0.951i)T \)
43 \( 1 + (0.309 + 0.951i)T \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (-0.309 + 0.951i)T \)
61 \( 1 + T \)
67 \( 1 - T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (0.809 - 0.587i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (-0.309 - 0.951i)T \)
89 \( 1 + (0.809 - 0.587i)T \)
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

−24.95413473587399563104451983991, −24.148451953917721655686696730, −23.173977118839434325855644328964, −22.40947643701577915318628341350, −21.60965447066044015861606864809, −20.399648305724943969648296509604, −19.62137825038382365608663334348, −18.79623783389941068473898192908, −17.60897377499116486104055956421, −16.91654636167700880087929218370, −15.847007288331840513930711744086, −15.269661560915964577801226577269, −14.70158734969284214875101346720, −13.86430785162375304098359289493, −12.29463383518883311457477632083, −11.33599520144011048294553601919, −10.29868526350548894759204784655, −9.27989722222033372769657151863, −8.3819889340199821741233069505, −7.90438727474414891958329421315, −6.55263995489783085314391032534, −5.36605300443172738868824846445, −4.37947730621558920723422667057, −3.790113334288435672854065400253, −1.863021063055435558166647661880, 0.4874011439604425237491430202, 1.58834073781633270880004440879, 2.84408547564125546650616410415, 3.887215136916617827399407941918, 4.8287796224884079412060362807, 6.69927972205050602450089909091, 7.558644549164203729006214217830, 8.479628765082612843551042286626, 9.016899317493071313629140340791, 10.699930226183209139729708487709, 11.403143279576563388895465825024, 12.00919966013040415065525875074, 12.99938474977425827152017075837, 13.966048666888099321208164850942, 14.568142448064101478585988118651, 16.07867699204585793406929876580, 17.23973319609509189449674128585, 17.89178686160433750751210174415, 18.797041782416999434418026897878, 19.64784095326137744092899901625, 20.04105639665365835378168606906, 20.88563042624416116377573711906, 22.16464901659186685563687950802, 22.90494107586772946462052848094, 24.013543525120347126708694274519

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