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Properties

Label 1-213-213.14-r0-0-0
Degree $1$
Conductor $213$
Sign $-0.387 + 0.921i$
Analytic cond. $0.989167$
Root an. cond. $0.989167$
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  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.309 + 0.951i)5-s + (0.809 + 0.587i)7-s + (−0.309 + 0.951i)8-s + (−0.809 + 0.587i)10-s + (−0.809 − 0.587i)11-s + (0.809 − 0.587i)13-s + (0.309 + 0.951i)14-s + (−0.809 + 0.587i)16-s + (−0.809 + 0.587i)17-s + (−0.809 − 0.587i)19-s − 20-s + (−0.309 − 0.951i)22-s + 23-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.309 + 0.951i)5-s + (0.809 + 0.587i)7-s + (−0.309 + 0.951i)8-s + (−0.809 + 0.587i)10-s + (−0.809 − 0.587i)11-s + (0.809 − 0.587i)13-s + (0.309 + 0.951i)14-s + (−0.809 + 0.587i)16-s + (−0.809 + 0.587i)17-s + (−0.809 − 0.587i)19-s − 20-s + (−0.309 − 0.951i)22-s + 23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 213 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.387 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.387 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(213\)    =    \(3 \cdot 71\)
Sign: $-0.387 + 0.921i$
Analytic conductor: \(0.989167\)
Root analytic conductor: \(0.989167\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{213} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 213,\ (0:\ ),\ -0.387 + 0.921i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9724552421 + 1.464266941i\)
\(L(\frac12)\) \(\approx\) \(0.9724552421 + 1.464266941i\)
\(L(1)\) \(\approx\) \(1.257468712 + 0.9042342031i\)
\(L(1)\) \(\approx\) \(1.257468712 + 0.9042342031i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
71 \( 1 \)
good2 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (-0.309 + 0.951i)T \)
7 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 + (-0.809 - 0.587i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.309 + 0.951i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + T \)
41 \( 1 + T \)
43 \( 1 + (0.309 - 0.951i)T \)
47 \( 1 + (-0.809 + 0.587i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (-0.309 + 0.951i)T \)
89 \( 1 + (-0.309 + 0.951i)T \)
97 \( 1 - T \)
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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

−26.4959635205400107636788915217, −25.02908368794321590606953622631, −24.27421518931232773565110103344, −23.372663727490443788599210220107, −22.92319870970990982054571915812, −21.15554851342983997185984967090, −20.95367550493910359733751953470, −20.09216631925106390308236718416, −19.03686543596571040124788433466, −17.89457664078189440506315160507, −16.658427035981748896523631663514, −15.63500828038515708704905496705, −14.72311433443854591491447473876, −13.4546543696545505437294612251, −12.97429002132887788781466917407, −11.67988531551358328563992737855, −11.02384216527775178400837626038, −9.79540770920958530096652246882, −8.54729720600531697819205355211, −7.31788518600381026462416533670, −5.86618516798062680681272186641, −4.58330878548896046234180667524, −4.20234515682517628559009251172, −2.36829343828179947364339820183, −1.10957310722456592943960326247, 2.36212595339828452169149132894, 3.34407598573173471455123755364, 4.68074309974696781984249793108, 5.80548728659607730940126857380, 6.76387569291070185955410747873, 7.98555152264706998268633068008, 8.67703382818388337418983857368, 10.92339404718938614661989056861, 11.122206318943115686608079106874, 12.625916676730574544072410396509, 13.50836112072397360164053776441, 14.653566523108854638099432175397, 15.27679649175870323190486003127, 16.03245088061790455581522877092, 17.52671270867231147276141447032, 18.16910440300502466287856281560, 19.32155515550474369252905136119, 20.75393675951822323415877111811, 21.51449336047018210925776477117, 22.27809046886727927347726838140, 23.36165513550649086591136174881, 23.92643375804140239164056046418, 25.028287666884663157090109970589, 25.90796649957028365193699044628, 26.6997561830793684027945826035

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