Properties

Label 2-208-1.1-c9-0-30
Degree $2$
Conductor $208$
Sign $-1$
Analytic cond. $107.127$
Root an. cond. $10.3502$
Motivic weight $9$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 75·3-s − 1.97e3·5-s + 1.01e4·7-s − 1.40e4·9-s − 1.88e4·11-s + 2.85e4·13-s + 1.48e5·15-s − 1.42e5·17-s − 8.33e4·19-s − 7.58e5·21-s + 5.36e5·23-s + 1.96e6·25-s + 2.53e6·27-s − 2.60e6·29-s + 2.21e6·31-s + 1.41e6·33-s − 2.00e7·35-s + 1.80e7·37-s − 2.14e6·39-s + 2.68e7·41-s + 4.22e7·43-s + 2.78e7·45-s − 3.59e7·47-s + 6.19e7·49-s + 1.06e7·51-s − 6.65e7·53-s + 3.73e7·55-s + ⋯
L(s)  = 1  − 0.534·3-s − 1.41·5-s + 1.59·7-s − 0.714·9-s − 0.388·11-s + 0.277·13-s + 0.757·15-s − 0.413·17-s − 0.146·19-s − 0.851·21-s + 0.399·23-s + 1.00·25-s + 0.916·27-s − 0.682·29-s + 0.430·31-s + 0.207·33-s − 2.25·35-s + 1.58·37-s − 0.148·39-s + 1.48·41-s + 1.88·43-s + 1.01·45-s − 1.07·47-s + 1.53·49-s + 0.221·51-s − 1.15·53-s + 0.549·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $-1$
Analytic conductor: \(107.127\)
Root analytic conductor: \(10.3502\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 208,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) 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 \( 1 - p^{4} T \)
good3 \( 1 + 25 p T + p^{9} T^{2} \)
5 \( 1 + 1979 T + p^{9} T^{2} \)
7 \( 1 - 1445 p T + p^{9} T^{2} \)
11 \( 1 + 18850 T + p^{9} T^{2} \)
17 \( 1 + 142403 T + p^{9} T^{2} \)
19 \( 1 + 83302 T + p^{9} T^{2} \)
23 \( 1 - 23328 p T + p^{9} T^{2} \)
29 \( 1 + 2600442 T + p^{9} T^{2} \)
31 \( 1 - 2214004 T + p^{9} T^{2} \)
37 \( 1 - 18099241 T + p^{9} T^{2} \)
41 \( 1 - 26812240 T + p^{9} T^{2} \)
43 \( 1 - 42253475 T + p^{9} T^{2} \)
47 \( 1 + 35914993 T + p^{9} T^{2} \)
53 \( 1 + 66514064 T + p^{9} T^{2} \)
59 \( 1 - 108164002 T + p^{9} T^{2} \)
61 \( 1 + 207449912 T + p^{9} T^{2} \)
67 \( 1 + 193015514 T + p^{9} T^{2} \)
71 \( 1 - 201833497 T + p^{9} T^{2} \)
73 \( 1 + 121628110 T + p^{9} T^{2} \)
79 \( 1 + 112871912 T + p^{9} T^{2} \)
83 \( 1 + 308254212 T + p^{9} T^{2} \)
89 \( 1 + 6374870 T + p^{9} T^{2} \)
97 \( 1 - 871266886 T + p^{9} T^{2} \)
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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

−10.87164736831376730501716357458, −9.014785991700314603661867409573, −8.040800780234869749936827364899, −7.56729365417839967163926607294, −6.03394843334441687020358376790, −4.88007654041583338667086649487, −4.14163118128551085213786457288, −2.65656802017290477783966125489, −1.08806307875591839704088806031, 0, 1.08806307875591839704088806031, 2.65656802017290477783966125489, 4.14163118128551085213786457288, 4.88007654041583338667086649487, 6.03394843334441687020358376790, 7.56729365417839967163926607294, 8.040800780234869749936827364899, 9.014785991700314603661867409573, 10.87164736831376730501716357458

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