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Properties

Label 2-208-13.12-c1-0-5
Degree $2$
Conductor $208$
Sign $0.554 + 0.832i$
Analytic cond. $1.66088$
Root an. cond. $1.28875$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 3i·5-s − 3i·7-s − 2·9-s + (2 + 3i)13-s − 3i·15-s + 3·17-s + 6i·19-s − 3i·21-s + 6·23-s − 4·25-s − 5·27-s − 9·35-s + 3i·37-s + (2 + 3i)39-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.34i·5-s − 1.13i·7-s − 0.666·9-s + (0.554 + 0.832i)13-s − 0.774i·15-s + 0.727·17-s + 1.37i·19-s − 0.654i·21-s + 1.25·23-s − 0.800·25-s − 0.962·27-s − 1.52·35-s + 0.493i·37-s + (0.320 + 0.480i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $0.554 + 0.832i$
Analytic conductor: \(1.66088\)
Root analytic conductor: \(1.28875\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 208,\ (\ :1/2),\ 0.554 + 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21299 - 0.649175i\)
\(L(\frac12)\) \(\approx\) \(1.21299 - 0.649175i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - 3i)T \)
good3 \( 1 - T + 3T^{2} \)
5 \( 1 + 3iT - 5T^{2} \)
7 \( 1 + 3iT - 7T^{2} \)
11 \( 1 - 11T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 - 6iT - 19T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 3iT - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + 3iT - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 6iT - 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 + 15iT - 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 + 6iT - 89T^{2} \)
97 \( 1 + 12iT - 97T^{2} \)
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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

−12.34696847050935546684804652172, −11.34321801490786955910351958186, −10.15131982791510938619243267869, −9.085027294343770777167117739192, −8.382982560575028275442200366894, −7.39812493034566573189772641080, −5.88295087585444615924522111830, −4.59575373214872018470438913342, −3.48628648044729452992003112336, −1.32750074296120482138943209608, 2.64144545797811667872857088370, 3.22516482017092836675604231153, 5.33610604815166923439313883019, 6.36392121518998361378249381242, 7.55056661700761708834136011260, 8.621273572444831610393501303142, 9.455261620798328003273195771539, 10.79841486861687876145581655720, 11.34913759877931000882715344370, 12.56591945348220648660290116662

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