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Properties

Label 1-293-293.11-r1-0-0
Degree $1$
Conductor $293$
Sign $-0.278 - 0.960i$
Analytic cond. $31.4872$
Root an. cond. $31.4872$
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  + (−0.785 + 0.618i)2-s + (0.493 − 0.869i)3-s + (0.234 − 0.972i)4-s + (0.668 + 0.744i)5-s + (0.150 + 0.988i)6-s + (0.758 − 0.651i)7-s + (0.417 + 0.908i)8-s + (−0.512 − 0.858i)9-s + (−0.985 − 0.171i)10-s + (−0.255 + 0.966i)11-s + (−0.729 − 0.683i)12-s + (−0.296 − 0.954i)13-s + (−0.192 + 0.981i)14-s + (0.976 − 0.213i)15-s + (−0.890 − 0.455i)16-s + (−0.618 − 0.785i)17-s + ⋯
L(s)  = 1  + (−0.785 + 0.618i)2-s + (0.493 − 0.869i)3-s + (0.234 − 0.972i)4-s + (0.668 + 0.744i)5-s + (0.150 + 0.988i)6-s + (0.758 − 0.651i)7-s + (0.417 + 0.908i)8-s + (−0.512 − 0.858i)9-s + (−0.985 − 0.171i)10-s + (−0.255 + 0.966i)11-s + (−0.729 − 0.683i)12-s + (−0.296 − 0.954i)13-s + (−0.192 + 0.981i)14-s + (0.976 − 0.213i)15-s + (−0.890 − 0.455i)16-s + (−0.618 − 0.785i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.278 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.278 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(293\)
Sign: $-0.278 - 0.960i$
Analytic conductor: \(31.4872\)
Root analytic conductor: \(31.4872\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{293} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 293,\ (1:\ ),\ -0.278 - 0.960i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7611982031 - 1.013659425i\)
\(L(\frac12)\) \(\approx\) \(0.7611982031 - 1.013659425i\)
\(L(1)\) \(\approx\) \(0.8995457906 - 0.1856224945i\)
\(L(1)\) \(\approx\) \(0.8995457906 - 0.1856224945i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad293 \( 1 \)
good2 \( 1 + (-0.785 + 0.618i)T \)
3 \( 1 + (0.493 - 0.869i)T \)
5 \( 1 + (0.668 + 0.744i)T \)
7 \( 1 + (0.758 - 0.651i)T \)
11 \( 1 + (-0.255 + 0.966i)T \)
13 \( 1 + (-0.296 - 0.954i)T \)
17 \( 1 + (-0.618 - 0.785i)T \)
19 \( 1 + (-0.213 - 0.976i)T \)
23 \( 1 + (-0.988 - 0.150i)T \)
29 \( 1 + (0.337 - 0.941i)T \)
31 \( 1 + (0.744 + 0.668i)T \)
37 \( 1 + (-0.584 - 0.811i)T \)
41 \( 1 + (0.530 - 0.847i)T \)
43 \( 1 + (-0.276 + 0.961i)T \)
47 \( 1 + (-0.917 - 0.397i)T \)
53 \( 1 + (0.966 + 0.255i)T \)
59 \( 1 + (-0.0645 + 0.997i)T \)
61 \( 1 + (-0.996 + 0.0859i)T \)
67 \( 1 + (0.991 - 0.128i)T \)
71 \( 1 + (-0.714 + 0.699i)T \)
73 \( 1 + (-0.234 - 0.972i)T \)
79 \( 1 + (-0.0859 - 0.996i)T \)
83 \( 1 + (-0.0215 - 0.999i)T \)
89 \( 1 + (0.0859 - 0.996i)T \)
97 \( 1 + (0.925 + 0.377i)T \)
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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

−25.77429584822870073814655787408, −24.775615946942156457731016708, −24.09253526466035167967959121731, −22.115396357506850690711955038518, −21.51440844782485226853572234677, −21.08745216442005086933514907460, −20.193589535872135944669698671723, −19.24546215782816156355673999056, −18.31346179279564225851345556033, −17.1970105080001365085303796503, −16.527749085169990342153284565789, −15.67795038620081588311490442299, −14.35870045708860344562707318737, −13.48382868346436123039623511647, −12.23800346333360960999155544953, −11.29544772885904874905488643577, −10.29810201156921559580073401634, −9.43688568791834592743541714894, −8.49129473494094848270654855367, −8.18650707350751195513167408159, −6.15891951267990875011664656666, −4.8861504428101914490893597556, −3.84998059961947820464672043856, −2.422275508093220891572432148678, −1.58613531912040343537219839372, 0.43528454908480355607671879388, 1.84092244718153552086714839427, 2.6543415269804707270884353443, 4.74596712546002446419720066900, 6.04171980358788523697507684021, 7.10978958120212723818768890645, 7.538244754406246205111108158410, 8.65272546447492590299601862894, 9.81584437434206623720236667150, 10.59948211577327681929252410950, 11.74822408323475852443945979045, 13.27394928674467023272784117070, 14.01598421038355855854999847966, 14.81053923150649496478787815651, 15.601694647601078106185962984693, 17.30188620132808299340439333616, 17.83670983129499790925344921411, 18.12654828189690311191190588953, 19.49516017999641764676912907999, 20.109936176872579284012443503, 21.04043751154171383519302643730, 22.71910184852350107611801823503, 23.287551232490579998835782084406, 24.58003111181973978822496246443, 24.81074800676415559509940396238

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