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Properties

Label 2-208-13.12-c3-0-1
Degree $2$
Conductor $208$
Sign $-0.966 + 0.257i$
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  − 1.18·3-s + 18.9i·5-s + 9.32i·7-s − 25.5·9-s − 39.8i·11-s + (−45.2 + 12.0i)13-s − 22.4i·15-s + 92.6·17-s − 128. i·19-s − 11.0i·21-s − 158.·23-s − 235.·25-s + 62.3·27-s − 126.·29-s + 189. i·31-s + ⋯
L(s)  = 1  − 0.227·3-s + 1.69i·5-s + 0.503i·7-s − 0.948·9-s − 1.09i·11-s + (−0.966 + 0.257i)13-s − 0.387i·15-s + 1.32·17-s − 1.55i·19-s − 0.114i·21-s − 1.44·23-s − 1.88·25-s + 0.444·27-s − 0.810·29-s + 1.09i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.257i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.966 + 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $-0.966 + 0.257i$
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: \((2,\ 208,\ (\ :3/2),\ -0.966 + 0.257i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0483132 - 0.368324i\)
\(L(\frac12)\) \(\approx\) \(0.0483132 - 0.368324i\)
\(L(\frac{5}{2})\) 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 \( 1 + (45.2 - 12.0i)T \)
good3 \( 1 + 1.18T + 27T^{2} \)
5 \( 1 - 18.9iT - 125T^{2} \)
7 \( 1 - 9.32iT - 343T^{2} \)
11 \( 1 + 39.8iT - 1.33e3T^{2} \)
17 \( 1 - 92.6T + 4.91e3T^{2} \)
19 \( 1 + 128. iT - 6.85e3T^{2} \)
23 \( 1 + 158.T + 1.21e4T^{2} \)
29 \( 1 + 126.T + 2.43e4T^{2} \)
31 \( 1 - 189. iT - 2.97e4T^{2} \)
37 \( 1 - 53.7iT - 5.06e4T^{2} \)
41 \( 1 + 136. iT - 6.89e4T^{2} \)
43 \( 1 + 518.T + 7.95e4T^{2} \)
47 \( 1 - 309. iT - 1.03e5T^{2} \)
53 \( 1 + 149.T + 1.48e5T^{2} \)
59 \( 1 - 74.3iT - 2.05e5T^{2} \)
61 \( 1 - 99.0T + 2.26e5T^{2} \)
67 \( 1 - 435. iT - 3.00e5T^{2} \)
71 \( 1 + 827. iT - 3.57e5T^{2} \)
73 \( 1 - 981. iT - 3.89e5T^{2} \)
79 \( 1 + 299.T + 4.93e5T^{2} \)
83 \( 1 - 169. iT - 5.71e5T^{2} \)
89 \( 1 - 265. iT - 7.04e5T^{2} \)
97 \( 1 - 88.9iT - 9.12e5T^{2} \)
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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

−12.09406336998752425879334061817, −11.45488024021042779438897610175, −10.67391702839429993293154421605, −9.684787153860334907528698922917, −8.418231434320176402838006310595, −7.27646751722284909279422610840, −6.27165870552560922904421465663, −5.37188875955058867281455158849, −3.35920608942285179723803830798, −2.55115939203711909803607562567, 0.15400089108686625085879355424, 1.75708252019026078384925661657, 3.89662394924030972982082814429, 5.06534466427331369800606175769, 5.84041969675711187688049539346, 7.65071644505894797551691694046, 8.244521649068312565815342740940, 9.610392800695043258434370253552, 10.12172491573643931457070648353, 11.91425912479137620400448154622

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