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Properties

Label 2-14e2-4.3-c0-0-0
Degree $2$
Conductor $196$
Sign $1$
Analytic cond. $0.0978167$
Root an. cond. $0.312756$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 8-s + 9-s + 16-s − 18-s − 25-s − 2·29-s − 32-s + 36-s − 2·37-s + 50-s + 2·53-s + 2·58-s + 64-s − 72-s + 2·74-s + 81-s − 100-s − 2·106-s − 2·109-s + 2·113-s − 2·116-s + ⋯
L(s)  = 1  − 2-s + 4-s − 8-s + 9-s + 16-s − 18-s − 25-s − 2·29-s − 32-s + 36-s − 2·37-s + 50-s + 2·53-s + 2·58-s + 64-s − 72-s + 2·74-s + 81-s − 100-s − 2·106-s − 2·109-s + 2·113-s − 2·116-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.0978167\)
Root analytic conductor: \(0.312756\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{196} (99, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4878156744\)
\(L(\frac12)\) \(\approx\) \(0.4878156744\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
7 \( 1 \)
good3 \( ( 1 - T )( 1 + T ) \)
5 \( 1 + T^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 + T )^{2} \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 + T )^{2} \)
41 \( 1 + T^{2} \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( 1 + T^{2} \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 + T^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( 1 + T^{2} \)
97 \( 1 + T^{2} \)
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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

−12.53191722332769884863651571813, −11.60687140681657786082542786270, −10.56991880068010727437427229026, −9.778317079932505908388739199955, −8.854075717860665745713986608863, −7.66109559082232720382507376407, −6.90375754611187260449823775175, −5.56686603511860954656868685423, −3.75433284828030179910406176303, −1.88434292211208747400285521709, 1.88434292211208747400285521709, 3.75433284828030179910406176303, 5.56686603511860954656868685423, 6.90375754611187260449823775175, 7.66109559082232720382507376407, 8.854075717860665745713986608863, 9.778317079932505908388739199955, 10.56991880068010727437427229026, 11.60687140681657786082542786270, 12.53191722332769884863651571813

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