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Properties

Label 2-208-13.3-c7-0-27
Degree $2$
Conductor $208$
Sign $0.664 - 0.747i$
Analytic cond. $64.9760$
Root an. cond. $8.06077$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (12.8 + 22.1i)3-s + 442.·5-s + (380. − 659. i)7-s + (765. − 1.32e3i)9-s + (3.05e3 + 5.29e3i)11-s + (409. + 7.91e3i)13-s + (5.66e3 + 9.81e3i)15-s + (−1.87e4 + 3.24e4i)17-s + (1.69e3 − 2.93e3i)19-s + 1.95e4·21-s + (−1.49e4 − 2.58e4i)23-s + 1.17e5·25-s + 9.52e4·27-s + (2.11e4 + 3.65e4i)29-s + 1.24e5·31-s + ⋯
L(s)  = 1  + (0.274 + 0.474i)3-s + 1.58·5-s + (0.419 − 0.726i)7-s + (0.349 − 0.605i)9-s + (0.692 + 1.19i)11-s + (0.0517 + 0.998i)13-s + (0.433 + 0.751i)15-s + (−0.926 + 1.60i)17-s + (0.0567 − 0.0983i)19-s + 0.459·21-s + (−0.255 − 0.443i)23-s + 1.50·25-s + 0.931·27-s + (0.160 + 0.278i)29-s + 0.747·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.664 - 0.747i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.664 - 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $0.664 - 0.747i$
Analytic conductor: \(64.9760\)
Root analytic conductor: \(8.06077\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 208,\ (\ :7/2),\ 0.664 - 0.747i)\)

Particular Values

\(L(4)\) \(\approx\) \(3.725636725\)
\(L(\frac12)\) \(\approx\) \(3.725636725\)
\(L(\frac{9}{2})\) 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 \( 1 + (-409. - 7.91e3i)T \)
good3 \( 1 + (-12.8 - 22.1i)T + (-1.09e3 + 1.89e3i)T^{2} \)
5 \( 1 - 442.T + 7.81e4T^{2} \)
7 \( 1 + (-380. + 659. i)T + (-4.11e5 - 7.13e5i)T^{2} \)
11 \( 1 + (-3.05e3 - 5.29e3i)T + (-9.74e6 + 1.68e7i)T^{2} \)
17 \( 1 + (1.87e4 - 3.24e4i)T + (-2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (-1.69e3 + 2.93e3i)T + (-4.46e8 - 7.74e8i)T^{2} \)
23 \( 1 + (1.49e4 + 2.58e4i)T + (-1.70e9 + 2.94e9i)T^{2} \)
29 \( 1 + (-2.11e4 - 3.65e4i)T + (-8.62e9 + 1.49e10i)T^{2} \)
31 \( 1 - 1.24e5T + 2.75e10T^{2} \)
37 \( 1 + (7.10e4 + 1.23e5i)T + (-4.74e10 + 8.22e10i)T^{2} \)
41 \( 1 + (-3.57e4 - 6.19e4i)T + (-9.73e10 + 1.68e11i)T^{2} \)
43 \( 1 + (6.38e3 - 1.10e4i)T + (-1.35e11 - 2.35e11i)T^{2} \)
47 \( 1 + 4.37e5T + 5.06e11T^{2} \)
53 \( 1 + 1.01e6T + 1.17e12T^{2} \)
59 \( 1 + (-8.79e5 + 1.52e6i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (-8.45e5 + 1.46e6i)T + (-1.57e12 - 2.72e12i)T^{2} \)
67 \( 1 + (-1.70e6 - 2.96e6i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 + (3.97e5 - 6.88e5i)T + (-4.54e12 - 7.87e12i)T^{2} \)
73 \( 1 - 3.45e6T + 1.10e13T^{2} \)
79 \( 1 + 6.86e6T + 1.92e13T^{2} \)
83 \( 1 - 8.04e6T + 2.71e13T^{2} \)
89 \( 1 + (-5.24e5 - 9.09e5i)T + (-2.21e13 + 3.83e13i)T^{2} \)
97 \( 1 + (5.62e5 - 9.75e5i)T + (-4.03e13 - 6.99e13i)T^{2} \)
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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

−10.96804609494046739417215929216, −10.02243936427835798816169923685, −9.510393717844608821142338394958, −8.559442966803491141884491805315, −6.85069937005433383465942283372, −6.32599174905702285076457112259, −4.70084033168923456851501731503, −3.95683453622646876611055229424, −2.09801078209084320074673691543, −1.36349135790126496746913337440, 0.924255951887077341990174260929, 2.05753096095787410180823584916, 2.90646231910220570291504791741, 4.96244559292228386561895883220, 5.78822390074793489819686509122, 6.76186092419084200535953791392, 8.125326672424758543480414600802, 8.991746228266619249701170244515, 9.889246760427674165572049802891, 10.96068406345758454974979632643

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