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Properties

Label 2-209-209.75-c0-0-0
Degree $2$
Conductor $209$
Sign $0.624 + 0.781i$
Analytic cond. $0.104304$
Root an. cond. $0.322962$
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)4-s + (−0.5 − 1.53i)5-s + (1.30 + 0.951i)7-s + (0.309 − 0.951i)9-s + (−0.809 + 0.587i)11-s + (0.309 + 0.951i)16-s + (0.190 + 0.587i)17-s + (−0.809 + 0.587i)19-s + (−0.5 + 1.53i)20-s + 0.618·23-s + (−1.30 + 0.951i)25-s + (−0.5 − 1.53i)28-s + (0.809 − 2.48i)35-s + (−0.809 + 0.587i)36-s + 0.618·43-s + 44-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)4-s + (−0.5 − 1.53i)5-s + (1.30 + 0.951i)7-s + (0.309 − 0.951i)9-s + (−0.809 + 0.587i)11-s + (0.309 + 0.951i)16-s + (0.190 + 0.587i)17-s + (−0.809 + 0.587i)19-s + (−0.5 + 1.53i)20-s + 0.618·23-s + (−1.30 + 0.951i)25-s + (−0.5 − 1.53i)28-s + (0.809 − 2.48i)35-s + (−0.809 + 0.587i)36-s + 0.618·43-s + 44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(209\)    =    \(11 \cdot 19\)
Sign: $0.624 + 0.781i$
Analytic conductor: \(0.104304\)
Root analytic conductor: \(0.322962\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{209} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 209,\ (\ :0),\ 0.624 + 0.781i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6196369510\)
\(L(\frac12)\) \(\approx\) \(0.6196369510\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + (0.809 - 0.587i)T \)
good2 \( 1 + (0.809 + 0.587i)T^{2} \)
3 \( 1 + (-0.309 + 0.951i)T^{2} \)
5 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
7 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
13 \( 1 + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 - 0.618T + T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 - 0.618T + T^{2} \)
47 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.809 + 0.587i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.55754108654927149206977030947, −11.77736733771123291824211224891, −10.44142081653990499334977380550, −9.234855821297590621975699829300, −8.631742236786995445683776810082, −7.87172071632455498761393167848, −5.84348882312015082626460189333, −4.94750417540238446597597982055, −4.23577991288248367378047185419, −1.52643915882307964783643299983, 2.76467273440989927742781032917, 4.12048781793688963050590855008, 5.11005384139615399801188318971, 7.08655780880078085318463662277, 7.69170128316693257239225794661, 8.441948033315951678875244942710, 10.14912427253826794641132597735, 10.92806231826180472754176426133, 11.40960441121386643588272252369, 12.97876538169174020162334207275

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