Properties

Label 2-208-13.12-c3-0-16
Degree $2$
Conductor $208$
Sign $0.429 + 0.903i$
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  + 8.86·3-s − 16.8i·5-s − 10.8i·7-s + 51.5·9-s + 35.1i·11-s + (−20.1 − 42.3i)13-s − 149. i·15-s − 30.3·17-s − 28.3i·19-s − 96.3i·21-s − 24.6·23-s − 159.·25-s + 218.·27-s + 290.·29-s + 219. i·31-s + ⋯
L(s)  = 1  + 1.70·3-s − 1.50i·5-s − 0.586i·7-s + 1.91·9-s + 0.964i·11-s + (−0.429 − 0.903i)13-s − 2.57i·15-s − 0.432·17-s − 0.342i·19-s − 1.00i·21-s − 0.223·23-s − 1.27·25-s + 1.55·27-s + 1.85·29-s + 1.27i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.429 + 0.903i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.429 + 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $0.429 + 0.903i$
Analytic conductor: \(12.2723\)
Root analytic conductor: \(3.50319\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 208,\ (\ :3/2),\ 0.429 + 0.903i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.54804 - 1.60956i\)
\(L(\frac12)\) \(\approx\) \(2.54804 - 1.60956i\)
\(L(\frac{5}{2})\) 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 \( 1 + (20.1 + 42.3i)T \)
good3 \( 1 - 8.86T + 27T^{2} \)
5 \( 1 + 16.8iT - 125T^{2} \)
7 \( 1 + 10.8iT - 343T^{2} \)
11 \( 1 - 35.1iT - 1.33e3T^{2} \)
17 \( 1 + 30.3T + 4.91e3T^{2} \)
19 \( 1 + 28.3iT - 6.85e3T^{2} \)
23 \( 1 + 24.6T + 1.21e4T^{2} \)
29 \( 1 - 290.T + 2.43e4T^{2} \)
31 \( 1 - 219. iT - 2.97e4T^{2} \)
37 \( 1 - 118. iT - 5.06e4T^{2} \)
41 \( 1 + 83.6iT - 6.89e4T^{2} \)
43 \( 1 - 293.T + 7.95e4T^{2} \)
47 \( 1 - 166. iT - 1.03e5T^{2} \)
53 \( 1 + 76.3T + 1.48e5T^{2} \)
59 \( 1 + 184. iT - 2.05e5T^{2} \)
61 \( 1 - 197.T + 2.26e5T^{2} \)
67 \( 1 - 321. iT - 3.00e5T^{2} \)
71 \( 1 + 368. iT - 3.57e5T^{2} \)
73 \( 1 - 843. iT - 3.89e5T^{2} \)
79 \( 1 + 184.T + 4.93e5T^{2} \)
83 \( 1 - 1.27e3iT - 5.71e5T^{2} \)
89 \( 1 + 1.36e3iT - 7.04e5T^{2} \)
97 \( 1 - 690. iT - 9.12e5T^{2} \)
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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

−12.28166378181333449004373345632, −10.36873810749932924517617637631, −9.563575750931252309748621213673, −8.683055014830795099706920841659, −8.034156911395797107479479920973, −7.03501727812601597695034663895, −4.96363584088814442342408417139, −4.13664624153263697809959941076, −2.63493039695462363420407603915, −1.16693116527910011872218892279, 2.25104309430081850129580706706, 2.95429020855017453737444134674, 4.08899427720723305158506816416, 6.18386359963952731515657926244, 7.18801518069643516720809942331, 8.177329205080405738672167929023, 9.064363722846416874891031294860, 9.980691910547106519053556663243, 11.01134945127010249505687596909, 12.11628772427321896485564612129

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