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Properties

Label 2-208-13.10-c3-0-7
Degree $2$
Conductor $208$
Sign $0.817 - 0.576i$
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  + (0.154 + 0.266i)3-s + 0.737i·5-s + (27.7 + 16.0i)7-s + (13.4 − 23.3i)9-s + (−28.2 + 16.3i)11-s + (−46.6 + 4.82i)13-s + (−0.196 + 0.113i)15-s + (20.0 − 34.7i)17-s + (93.5 + 54.0i)19-s + 9.88i·21-s + (108. + 187. i)23-s + 124.·25-s + 16.6·27-s + (107. + 186. i)29-s − 220. i·31-s + ⋯
L(s)  = 1  + (0.0296 + 0.0513i)3-s + 0.0659i·5-s + (1.50 + 0.866i)7-s + (0.498 − 0.862i)9-s + (−0.774 + 0.446i)11-s + (−0.994 + 0.103i)13-s + (−0.00338 + 0.00195i)15-s + (0.286 − 0.496i)17-s + (1.13 + 0.652i)19-s + 0.102i·21-s + (0.980 + 1.69i)23-s + 0.995·25-s + 0.118·27-s + (0.689 + 1.19i)29-s − 1.27i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 - 0.576i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.817 - 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $0.817 - 0.576i$
Analytic conductor: \(12.2723\)
Root analytic conductor: \(3.50319\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 208,\ (\ :3/2),\ 0.817 - 0.576i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.94515 + 0.616658i\)
\(L(\frac12)\) \(\approx\) \(1.94515 + 0.616658i\)
\(L(\frac{5}{2})\) 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 \( 1 + (46.6 - 4.82i)T \)
good3 \( 1 + (-0.154 - 0.266i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 - 0.737iT - 125T^{2} \)
7 \( 1 + (-27.7 - 16.0i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (28.2 - 16.3i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-20.0 + 34.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-93.5 - 54.0i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-108. - 187. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-107. - 186. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 220. iT - 2.97e4T^{2} \)
37 \( 1 + (-192. + 111. i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (316. - 182. i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (40.9 - 70.9i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 92.6iT - 1.03e5T^{2} \)
53 \( 1 - 72.5T + 1.48e5T^{2} \)
59 \( 1 + (376. + 217. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-173. + 299. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (490. - 282. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-221. - 127. i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 714. iT - 3.89e5T^{2} \)
79 \( 1 + 365.T + 4.93e5T^{2} \)
83 \( 1 - 467. iT - 5.71e5T^{2} \)
89 \( 1 + (367. - 211. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (513. + 296. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−11.95625934965959902194068504370, −11.27944612117599466536145244446, −9.934952385211557987308292623351, −9.167121380414931498125026059454, −7.899315167897499211244291132362, −7.16113744094399188650915333141, −5.44559174203794712290196104945, −4.78516642307624430507915452302, −2.98549799589032470607644735690, −1.44949277680533009309449254510, 1.03671191688521282038846447423, 2.65007515722773572658711272106, 4.64747967573805635023283114754, 5.07222193599871064387010412997, 7.00602994492875810971508416870, 7.80002471251272384247407660821, 8.590515722604130139267976656662, 10.29199127388288812400516784997, 10.66904179978900494570084682726, 11.74672194140532114624166770799

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