Properties

Label 1-307-307.216-r0-0-0
Degree $1$
Conductor $307$
Sign $-0.0543 - 0.998i$
Analytic cond. $1.42570$
Root an. cond. $1.42570$
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  + (−0.602 − 0.798i)2-s + (0.445 + 0.895i)3-s + (−0.273 + 0.961i)4-s + (−0.273 − 0.961i)5-s + (0.445 − 0.895i)6-s + (0.739 − 0.673i)7-s + (0.932 − 0.361i)8-s + (−0.602 + 0.798i)9-s + (−0.602 + 0.798i)10-s + (0.445 − 0.895i)11-s + (−0.982 + 0.183i)12-s + (−0.850 − 0.526i)13-s + (−0.982 − 0.183i)14-s + (0.739 − 0.673i)15-s + (−0.850 − 0.526i)16-s + 17-s + ⋯
L(s)  = 1  + (−0.602 − 0.798i)2-s + (0.445 + 0.895i)3-s + (−0.273 + 0.961i)4-s + (−0.273 − 0.961i)5-s + (0.445 − 0.895i)6-s + (0.739 − 0.673i)7-s + (0.932 − 0.361i)8-s + (−0.602 + 0.798i)9-s + (−0.602 + 0.798i)10-s + (0.445 − 0.895i)11-s + (−0.982 + 0.183i)12-s + (−0.850 − 0.526i)13-s + (−0.982 − 0.183i)14-s + (0.739 − 0.673i)15-s + (−0.850 − 0.526i)16-s + 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 307 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0543 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 307 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0543 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(307\)
Sign: $-0.0543 - 0.998i$
Analytic conductor: \(1.42570\)
Root analytic conductor: \(1.42570\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{307} (216, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 307,\ (0:\ ),\ -0.0543 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6468810410 - 0.6830207989i\)
\(L(\frac12)\) \(\approx\) \(0.6468810410 - 0.6830207989i\)
\(L(1)\) \(\approx\) \(0.7993616729 - 0.3403705201i\)
\(L(1)\) \(\approx\) \(0.7993616729 - 0.3403705201i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad307 \( 1 \)
good2 \( 1 + (-0.602 - 0.798i)T \)
3 \( 1 + (0.445 + 0.895i)T \)
5 \( 1 + (-0.273 - 0.961i)T \)
7 \( 1 + (0.739 - 0.673i)T \)
11 \( 1 + (0.445 - 0.895i)T \)
13 \( 1 + (-0.850 - 0.526i)T \)
17 \( 1 + T \)
19 \( 1 + (-0.850 - 0.526i)T \)
23 \( 1 + (-0.602 + 0.798i)T \)
29 \( 1 + (-0.602 - 0.798i)T \)
31 \( 1 + (0.445 - 0.895i)T \)
37 \( 1 + (0.739 - 0.673i)T \)
41 \( 1 + (0.739 + 0.673i)T \)
43 \( 1 + (0.932 - 0.361i)T \)
47 \( 1 + (-0.850 - 0.526i)T \)
53 \( 1 + T \)
59 \( 1 + (0.0922 - 0.995i)T \)
61 \( 1 + (0.932 - 0.361i)T \)
67 \( 1 + (-0.982 + 0.183i)T \)
71 \( 1 + (0.932 + 0.361i)T \)
73 \( 1 + (-0.850 - 0.526i)T \)
79 \( 1 + (0.932 + 0.361i)T \)
83 \( 1 + (0.739 + 0.673i)T \)
89 \( 1 + (-0.982 + 0.183i)T \)
97 \( 1 + (-0.850 + 0.526i)T \)
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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

−25.558811073817903706344734302995, −24.74120929621905862598052164400, −23.95276292793684697578902567288, −23.13666600606351821325333657295, −22.2613507329014275221699880720, −20.86219406734733310913628208135, −19.61478078521979247685254561044, −19.0559811561901380278577579659, −18.21743289330725538777374868737, −17.67812172207705243715832967404, −16.58781616979892530087002144903, −15.12205052946633689451884502447, −14.556514570264038074636232314304, −14.22468556368501827836173050819, −12.50700385999566310370760945449, −11.71100600910469691733465797568, −10.417701783102105450249663833341, −9.35117039006603904839917519416, −8.28445481448443748324870689085, −7.51140130917739914105242503958, −6.76158363283146234344755342792, −5.797475803647989735841067411952, −4.33280395819932736808769767808, −2.52707690056243161375230649155, −1.60615632495339174946499956179, 0.74032282725203399044678517205, 2.25129535931112816232038964072, 3.6818309496624504896280745578, 4.33041141697624472581979637899, 5.44222438799259663303384431766, 7.74830601573594469960679994901, 8.14178423046723870375883356785, 9.250461808550199533947016700486, 9.97106055872959844324311853734, 11.07353162792265846170485396100, 11.75100520044044166041480779514, 13.02280789035764032761565783455, 13.919179671410132789759379102607, 15.02812034148968106048307341645, 16.31832640179981712012818855691, 16.89970560237061293151435245387, 17.58610661896190249794954214647, 19.25546799190585662485260036855, 19.69224109902231878835335071740, 20.60109454916574001177056885336, 21.19740663150723197942552352100, 21.92142896919284129700482039973, 23.118517687387177771190845667284, 24.33483633501771991991189992393, 25.20774543053787086902997389563

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