Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·5-s − 2·7-s − 2·9-s + 2·11-s − 7·13-s + 2·15-s − 3·17-s + 5·19-s + 2·21-s − 3·23-s − 25-s + 5·27-s + 9·29-s − 8·31-s − 2·33-s + 4·35-s − 3·37-s + 7·39-s + 2·41-s + 4·43-s + 4·45-s + 10·47-s − 3·49-s + 3·51-s + 53-s − 4·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s − 0.755·7-s − 2/3·9-s + 0.603·11-s − 1.94·13-s + 0.516·15-s − 0.727·17-s + 1.14·19-s + 0.436·21-s − 0.625·23-s − 1/5·25-s + 0.962·27-s + 1.67·29-s − 1.43·31-s − 0.348·33-s + 0.676·35-s − 0.493·37-s + 1.12·39-s + 0.312·41-s + 0.609·43-s + 0.596·45-s + 1.45·47-s − 3/7·49-s + 0.420·51-s + 0.137·53-s − 0.539·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 212 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(212\)    =    \(2^{2} \cdot 53\)
Sign: $-1$
Analytic conductor: \(1.69282\)
Root analytic conductor: \(1.30108\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 212,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
53 \( 1 - T \)
good3 \( 1 + T + p T^{2} \) 1.3.b
5 \( 1 + 2 T + p T^{2} \) 1.5.c
7 \( 1 + 2 T + p T^{2} \) 1.7.c
11 \( 1 - 2 T + p T^{2} \) 1.11.ac
13 \( 1 + 7 T + p T^{2} \) 1.13.h
17 \( 1 + 3 T + p T^{2} \) 1.17.d
19 \( 1 - 5 T + p T^{2} \) 1.19.af
23 \( 1 + 3 T + p T^{2} \) 1.23.d
29 \( 1 - 9 T + p T^{2} \) 1.29.aj
31 \( 1 + 8 T + p T^{2} \) 1.31.i
37 \( 1 + 3 T + p T^{2} \) 1.37.d
41 \( 1 - 2 T + p T^{2} \) 1.41.ac
43 \( 1 - 4 T + p T^{2} \) 1.43.ae
47 \( 1 - 10 T + p T^{2} \) 1.47.ak
59 \( 1 + 2 T + p T^{2} \) 1.59.c
61 \( 1 + 10 T + p T^{2} \) 1.61.k
67 \( 1 - 4 T + p T^{2} \) 1.67.ae
71 \( 1 + 9 T + p T^{2} \) 1.71.j
73 \( 1 + 6 T + p T^{2} \) 1.73.g
79 \( 1 - 5 T + p T^{2} \) 1.79.af
83 \( 1 + 11 T + p T^{2} \) 1.83.l
89 \( 1 + 10 T + p T^{2} \) 1.89.k
97 \( 1 + 3 T + p T^{2} \) 1.97.d
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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

−12.13335372223286490790910750723, −11.08240569521732123789081761888, −9.930666925609079478275316289925, −8.994138611563358629998968833701, −7.65858768213842096979929548729, −6.79300825375205190228276253732, −5.54605997486662159198728085456, −4.31696655629561992549529664137, −2.87037683964475092833726904229, 0, 2.87037683964475092833726904229, 4.31696655629561992549529664137, 5.54605997486662159198728085456, 6.79300825375205190228276253732, 7.65858768213842096979929548729, 8.994138611563358629998968833701, 9.930666925609079478275316289925, 11.08240569521732123789081761888, 12.13335372223286490790910750723

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