Properties

Label 2-208-1.1-c7-0-23
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  + 87·3-s + 321·5-s + 181·7-s + 5.38e3·9-s − 7.78e3·11-s + 2.19e3·13-s + 2.79e4·15-s + 9.06e3·17-s + 3.71e4·19-s + 1.57e4·21-s − 1.90e4·23-s + 2.49e4·25-s + 2.77e5·27-s + 1.74e5·29-s − 2.90e4·31-s − 6.77e5·33-s + 5.81e4·35-s + 3.23e5·37-s + 1.91e5·39-s + 7.95e5·41-s + 3.14e5·43-s + 1.72e6·45-s + 4.47e5·47-s − 7.90e5·49-s + 7.89e5·51-s − 1.46e6·53-s − 2.49e6·55-s + ⋯
L(s)  = 1  + 1.86·3-s + 1.14·5-s + 0.199·7-s + 2.46·9-s − 1.76·11-s + 0.277·13-s + 2.13·15-s + 0.447·17-s + 1.24·19-s + 0.371·21-s − 0.325·23-s + 0.318·25-s + 2.71·27-s + 1.33·29-s − 0.174·31-s − 3.27·33-s + 0.229·35-s + 1.05·37-s + 0.515·39-s + 1.80·41-s + 0.602·43-s + 2.82·45-s + 0.628·47-s − 0.960·49-s + 0.832·51-s − 1.35·53-s − 2.02·55-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

\(L(4)\) \(\approx\) \(5.604873967\)
\(L(\frac12)\) \(\approx\) \(5.604873967\)
\(L(\frac{9}{2})\) 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 \( 1 - 29 p T + p^{7} T^{2} \)
5 \( 1 - 321 T + p^{7} T^{2} \)
7 \( 1 - 181 T + p^{7} T^{2} \)
11 \( 1 + 7782 T + p^{7} T^{2} \)
17 \( 1 - 9069 T + p^{7} T^{2} \)
19 \( 1 - 37150 T + p^{7} T^{2} \)
23 \( 1 + 19008 T + p^{7} T^{2} \)
29 \( 1 - 174750 T + p^{7} T^{2} \)
31 \( 1 + 29012 T + p^{7} T^{2} \)
37 \( 1 - 323669 T + p^{7} T^{2} \)
41 \( 1 - 795312 T + p^{7} T^{2} \)
43 \( 1 - 314137 T + p^{7} T^{2} \)
47 \( 1 - 447441 T + p^{7} T^{2} \)
53 \( 1 + 1469232 T + p^{7} T^{2} \)
59 \( 1 + 1627770 T + p^{7} T^{2} \)
61 \( 1 + 2399608 T + p^{7} T^{2} \)
67 \( 1 - 64066 T + p^{7} T^{2} \)
71 \( 1 - 322383 T + p^{7} T^{2} \)
73 \( 1 + 4454782 T + p^{7} T^{2} \)
79 \( 1 + 753560 T + p^{7} T^{2} \)
83 \( 1 - 1219092 T + p^{7} T^{2} \)
89 \( 1 - 3390330 T + p^{7} T^{2} \)
97 \( 1 - 1628774 T + p^{7} T^{2} \)
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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

−10.68820960374450900131546695534, −9.843176487017801747008106428227, −9.238614412335634051242095855762, −8.036763623780548667019261069780, −7.55678578877642434648576875262, −5.91748463179311430334484546146, −4.64533484768492731433238912075, −3.06370988367895521500575277053, −2.45475365795334637302684600950, −1.29682391417936833652142650118, 1.29682391417936833652142650118, 2.45475365795334637302684600950, 3.06370988367895521500575277053, 4.64533484768492731433238912075, 5.91748463179311430334484546146, 7.55678578877642434648576875262, 8.036763623780548667019261069780, 9.238614412335634051242095855762, 9.843176487017801747008106428227, 10.68820960374450900131546695534

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