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Properties

Label 2-208-13.12-c3-0-17
Degree $2$
Conductor $208$
Sign $-0.896 - 0.443i$
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  − 7·3-s − 20.8i·5-s − 20.8i·7-s + 22·9-s + (42 + 20.8i)13-s + 145. i·15-s − 77·17-s − 124. i·19-s + 145. i·21-s − 34·23-s − 308.·25-s + 35·27-s − 64·29-s + 166. i·31-s − 433.·35-s + ⋯
L(s)  = 1  − 1.34·3-s − 1.86i·5-s − 1.12i·7-s + 0.814·9-s + (0.896 + 0.443i)13-s + 2.50i·15-s − 1.09·17-s − 1.50i·19-s + 1.51i·21-s − 0.308·23-s − 2.46·25-s + 0.249·27-s − 0.409·29-s + 0.964i·31-s − 2.09·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 - 0.443i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $-0.896 - 0.443i$
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: \((2,\ 208,\ (\ :3/2),\ -0.896 - 0.443i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.115703 + 0.494161i\)
\(L(\frac12)\) \(\approx\) \(0.115703 + 0.494161i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-42 - 20.8i)T \)
good3 \( 1 + 7T + 27T^{2} \)
5 \( 1 + 20.8iT - 125T^{2} \)
7 \( 1 + 20.8iT - 343T^{2} \)
11 \( 1 - 1.33e3T^{2} \)
17 \( 1 + 77T + 4.91e3T^{2} \)
19 \( 1 + 124. iT - 6.85e3T^{2} \)
23 \( 1 + 34T + 1.21e4T^{2} \)
29 \( 1 + 64T + 2.43e4T^{2} \)
31 \( 1 - 166. iT - 2.97e4T^{2} \)
37 \( 1 + 145. iT - 5.06e4T^{2} \)
41 \( 1 - 332. iT - 6.89e4T^{2} \)
43 \( 1 + 55T + 7.95e4T^{2} \)
47 \( 1 - 312. iT - 1.03e5T^{2} \)
53 \( 1 - 594T + 1.48e5T^{2} \)
59 \( 1 - 541. iT - 2.05e5T^{2} \)
61 \( 1 + 280T + 2.26e5T^{2} \)
67 \( 1 + 582. iT - 3.00e5T^{2} \)
71 \( 1 + 436. iT - 3.57e5T^{2} \)
73 \( 1 + 457. iT - 3.89e5T^{2} \)
79 \( 1 + 594T + 4.93e5T^{2} \)
83 \( 1 - 457. iT - 5.71e5T^{2} \)
89 \( 1 - 124. iT - 7.04e5T^{2} \)
97 \( 1 + 749. iT - 9.12e5T^{2} \)
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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

−11.37006503642355365604845493130, −10.70788997723636700976379984868, −9.337669805033067042650174978350, −8.523789247104732493101549988310, −7.08603000825782513744683734413, −5.99221761425433398155400484614, −4.83271252307184842164832843828, −4.25894643085392051163848023314, −1.26406479307688217946550243638, −0.28203555404151220559021373046, 2.27438401536724150889028177596, 3.74401449325069823171286392034, 5.66912607785829097729541046074, 6.10398353536083453178034263201, 7.06191549947009451816106299290, 8.440264712443083404572134960942, 10.00520187920398927015361532801, 10.73597473051252915076045412388, 11.45478153444543027912481127622, 12.07395023198675769853667816971

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