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Properties

Label 2-208-13.3-c3-0-9
Degree $2$
Conductor $208$
Sign $-0.262 - 0.964i$
Analytic cond. $12.2723$
Root an. cond. $3.50319$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.34 + 7.52i)3-s + 2.80·5-s + (4.78 − 8.28i)7-s + (−24.2 + 41.9i)9-s + (19.7 + 34.1i)11-s + (40.5 + 23.5i)13-s + (12.1 + 21.1i)15-s + (−1.00 + 1.74i)17-s + (−30.0 + 52.1i)19-s + 83.0·21-s + (2.23 + 3.87i)23-s − 117.·25-s − 186.·27-s + (−70.3 − 121. i)29-s − 136.·31-s + ⋯
L(s)  = 1  + (0.835 + 1.44i)3-s + 0.251·5-s + (0.258 − 0.447i)7-s + (−0.896 + 1.55i)9-s + (0.540 + 0.935i)11-s + (0.864 + 0.502i)13-s + (0.209 + 0.363i)15-s + (−0.0143 + 0.0248i)17-s + (−0.363 + 0.629i)19-s + 0.862·21-s + (0.0202 + 0.0350i)23-s − 0.936·25-s − 1.32·27-s + (−0.450 − 0.780i)29-s − 0.788·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.262 - 0.964i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.262 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $-0.262 - 0.964i$
Analytic conductor: \(12.2723\)
Root analytic conductor: \(3.50319\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 208,\ (\ :3/2),\ -0.262 - 0.964i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.47808 + 1.93352i\)
\(L(\frac12)\) \(\approx\) \(1.47808 + 1.93352i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (-40.5 - 23.5i)T \)
good3 \( 1 + (-4.34 - 7.52i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 - 2.80T + 125T^{2} \)
7 \( 1 + (-4.78 + 8.28i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-19.7 - 34.1i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (1.00 - 1.74i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (30.0 - 52.1i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-2.23 - 3.87i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (70.3 + 121. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 136.T + 2.97e4T^{2} \)
37 \( 1 + (-92.8 - 160. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (155. + 268. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-213. + 370. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 258.T + 1.03e5T^{2} \)
53 \( 1 - 612.T + 1.48e5T^{2} \)
59 \( 1 + (258. - 448. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-80.6 + 139. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (24.9 + 43.2i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-139. + 242. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 467.T + 3.89e5T^{2} \)
79 \( 1 + 37.5T + 4.93e5T^{2} \)
83 \( 1 - 76.1T + 5.71e5T^{2} \)
89 \( 1 + (101. + 175. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-587. + 1.01e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−12.09502329349812990372287326816, −10.92175992102991177065429632216, −10.13881571504786322358225037986, −9.343842474169628529287389933033, −8.549563183363349354025217092854, −7.32544761464690183741227703011, −5.75482459267007670211840252554, −4.30876939538128267703829337624, −3.78211737237889501922796414587, −2.02052639749430810660029952224, 1.01468114545674974357283116160, 2.29543924734099794720164215931, 3.56388907414323892461458756641, 5.68445219947389518289336802131, 6.55953864886868256421572928772, 7.70886350565908556722100227717, 8.563067176028883767023670541777, 9.227339017378569954506923538223, 10.92451008787987678810717287185, 11.80493011462981429459850126705

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