Properties

Label 2-208-1.1-c7-0-22
Degree $2$
Conductor $208$
Sign $1$
Analytic cond. $64.9760$
Root an. cond. $8.06077$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 39·3-s + 385·5-s + 293·7-s − 666·9-s + 5.40e3·11-s + 2.19e3·13-s + 1.50e4·15-s − 2.10e4·17-s + 2.73e4·19-s + 1.14e4·21-s + 6.30e4·23-s + 7.01e4·25-s − 1.11e5·27-s + 1.22e5·29-s + 2.08e5·31-s + 2.10e5·33-s + 1.12e5·35-s − 4.42e5·37-s + 8.56e4·39-s + 5.80e4·41-s + 2.02e5·43-s − 2.56e5·45-s − 5.88e5·47-s − 7.37e5·49-s − 8.19e5·51-s + 1.68e6·53-s + 2.07e6·55-s + ⋯
L(s)  = 1  + 0.833·3-s + 1.37·5-s + 0.322·7-s − 0.304·9-s + 1.22·11-s + 0.277·13-s + 1.14·15-s − 1.03·17-s + 0.913·19-s + 0.269·21-s + 1.08·23-s + 0.897·25-s − 1.08·27-s + 0.930·29-s + 1.25·31-s + 1.02·33-s + 0.444·35-s − 1.43·37-s + 0.231·39-s + 0.131·41-s + 0.387·43-s − 0.419·45-s − 0.826·47-s − 0.895·49-s − 0.864·51-s + 1.55·53-s + 1.68·55-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $1$
Analytic conductor: \(64.9760\)
Root analytic conductor: \(8.06077\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 208,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(4.300219298\)
\(L(\frac12)\) \(\approx\) \(4.300219298\)
\(L(\frac{9}{2})\) 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 \( 1 - p^{3} T \)
good3 \( 1 - 13 p T + p^{7} T^{2} \)
5 \( 1 - 77 p T + p^{7} T^{2} \)
7 \( 1 - 293 T + p^{7} T^{2} \)
11 \( 1 - 5402 T + p^{7} T^{2} \)
17 \( 1 + 21011 T + p^{7} T^{2} \)
19 \( 1 - 27326 T + p^{7} T^{2} \)
23 \( 1 - 63072 T + p^{7} T^{2} \)
29 \( 1 - 122238 T + p^{7} T^{2} \)
31 \( 1 - 208396 T + p^{7} T^{2} \)
37 \( 1 + 442379 T + p^{7} T^{2} \)
41 \( 1 - 58000 T + p^{7} T^{2} \)
43 \( 1 - 202025 T + p^{7} T^{2} \)
47 \( 1 + 588511 T + p^{7} T^{2} \)
53 \( 1 - 1684336 T + p^{7} T^{2} \)
59 \( 1 - 442630 T + p^{7} T^{2} \)
61 \( 1 + 1083608 T + p^{7} T^{2} \)
67 \( 1 + 3443486 T + p^{7} T^{2} \)
71 \( 1 + 2084705 T + p^{7} T^{2} \)
73 \( 1 - 5937890 T + p^{7} T^{2} \)
79 \( 1 - 6609256 T + p^{7} T^{2} \)
83 \( 1 - 142740 T + p^{7} T^{2} \)
89 \( 1 + 6985286 T + p^{7} T^{2} \)
97 \( 1 + 200762 T + p^{7} T^{2} \)
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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

−11.06241730763883674197879983820, −9.869458638484497953213639069978, −9.081535383735702566175584962139, −8.470515076948920233092280204226, −6.94350983123506652418332940873, −6.03000969731927806460238535384, −4.80332465730645020070286328846, −3.29541025988372532650710634101, −2.19961843821742036725736426108, −1.16158263783275603643873037832, 1.16158263783275603643873037832, 2.19961843821742036725736426108, 3.29541025988372532650710634101, 4.80332465730645020070286328846, 6.03000969731927806460238535384, 6.94350983123506652418332940873, 8.470515076948920233092280204226, 9.081535383735702566175584962139, 9.869458638484497953213639069978, 11.06241730763883674197879983820

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