Properties

Label 2-208-13.12-c3-0-11
Degree $2$
Conductor $208$
Sign $-0.408 + 0.912i$
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  − 9.27·3-s − 2.27i·5-s + 28.5i·7-s + 59.0·9-s + 0.604i·11-s + (−19.1 + 42.7i)13-s + 21.1i·15-s − 41.0·17-s − 129. i·19-s − 264. i·21-s − 73.8·23-s + 119.·25-s − 297.·27-s − 214.·29-s − 126. i·31-s + ⋯
L(s)  = 1  − 1.78·3-s − 0.203i·5-s + 1.54i·7-s + 2.18·9-s + 0.0165i·11-s + (−0.408 + 0.912i)13-s + 0.364i·15-s − 0.585·17-s − 1.56i·19-s − 2.75i·21-s − 0.669·23-s + 0.958·25-s − 2.11·27-s − 1.37·29-s − 0.731i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.408 + 0.912i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.408 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $-0.408 + 0.912i$
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: \((2,\ 208,\ (\ :3/2),\ -0.408 + 0.912i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.131453 - 0.202881i\)
\(L(\frac12)\) \(\approx\) \(0.131453 - 0.202881i\)
\(L(\frac{5}{2})\) 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 \( 1 + (19.1 - 42.7i)T \)
good3 \( 1 + 9.27T + 27T^{2} \)
5 \( 1 + 2.27iT - 125T^{2} \)
7 \( 1 - 28.5iT - 343T^{2} \)
11 \( 1 - 0.604iT - 1.33e3T^{2} \)
17 \( 1 + 41.0T + 4.91e3T^{2} \)
19 \( 1 + 129. iT - 6.85e3T^{2} \)
23 \( 1 + 73.8T + 1.21e4T^{2} \)
29 \( 1 + 214.T + 2.43e4T^{2} \)
31 \( 1 + 126. iT - 2.97e4T^{2} \)
37 \( 1 + 309. iT - 5.06e4T^{2} \)
41 \( 1 - 88.1iT - 6.89e4T^{2} \)
43 \( 1 - 245.T + 7.95e4T^{2} \)
47 \( 1 + 68.5iT - 1.03e5T^{2} \)
53 \( 1 + 613.T + 1.48e5T^{2} \)
59 \( 1 + 840. iT - 2.05e5T^{2} \)
61 \( 1 - 587.T + 2.26e5T^{2} \)
67 \( 1 - 606. iT - 3.00e5T^{2} \)
71 \( 1 - 507. iT - 3.57e5T^{2} \)
73 \( 1 + 177. iT - 3.89e5T^{2} \)
79 \( 1 + 143.T + 4.93e5T^{2} \)
83 \( 1 + 773. iT - 5.71e5T^{2} \)
89 \( 1 + 1.40e3iT - 7.04e5T^{2} \)
97 \( 1 + 468. iT - 9.12e5T^{2} \)
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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

−11.38780975505475394941574680157, −11.22676560198704009018203719863, −9.664648109068732257490754821661, −8.899092712382486641174741650860, −7.17812175467414869554763036877, −6.22102280932735808312945492740, −5.36106507067047624549403922935, −4.48734616053658360160733077715, −2.12791210160171793690261668254, −0.14719119400874533529060132351, 1.17618795706115584360575473358, 3.82444633416363208527805595583, 4.92646577173430444123346638789, 6.02892115854147508892689535212, 6.97681783605606831019569902594, 7.84123801734945184154140738735, 9.865134538280100747782586281915, 10.54384382562525103928261170633, 11.04956862941164745647764537062, 12.20478920050425781136745853736

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