Properties

Degree $2$
Conductor $1$
Sign $-1$
Motivic weight $105$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.72e15·2-s − 1.28e25·3-s − 3.75e31·4-s − 7.91e36·5-s − 2.21e40·6-s + 2.76e44·7-s − 1.34e47·8-s + 4.01e49·9-s − 1.36e52·10-s − 6.07e54·11-s + 4.83e56·12-s + 2.29e58·13-s + 4.76e59·14-s + 1.01e62·15-s + 1.29e63·16-s − 1.76e64·17-s + 6.92e64·18-s + 1.67e67·19-s + 2.97e68·20-s − 3.55e69·21-s − 1.04e70·22-s + 4.70e71·23-s + 1.73e72·24-s + 3.79e73·25-s + 3.95e73·26-s + 1.09e75·27-s − 1.03e76·28-s + ⋯
L(s)  = 1  + 0.270·2-s − 1.14·3-s − 0.926·4-s − 1.59·5-s − 0.311·6-s + 1.18·7-s − 0.521·8-s + 0.320·9-s − 0.431·10-s − 1.28·11-s + 1.06·12-s + 0.756·13-s + 0.320·14-s + 1.83·15-s + 0.785·16-s − 0.445·17-s + 0.0868·18-s + 1.22·19-s + 1.47·20-s − 1.36·21-s − 0.348·22-s + 1.52·23-s + 0.599·24-s + 1.54·25-s + 0.204·26-s + 0.780·27-s − 1.09·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(106-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+52.5) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $-1$
Motivic weight: \(105\)
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1,\ (\ :105/2),\ -1)\)

Particular Values

\(L(53)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{107}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 - 1.72e15T + 4.05e31T^{2} \)
3 \( 1 + 1.28e25T + 1.25e50T^{2} \)
5 \( 1 + 7.91e36T + 2.46e73T^{2} \)
7 \( 1 - 2.76e44T + 5.43e88T^{2} \)
11 \( 1 + 6.07e54T + 2.21e109T^{2} \)
13 \( 1 - 2.29e58T + 9.20e116T^{2} \)
17 \( 1 + 1.76e64T + 1.57e129T^{2} \)
19 \( 1 - 1.67e67T + 1.85e134T^{2} \)
23 \( 1 - 4.70e71T + 9.58e142T^{2} \)
29 \( 1 + 2.66e76T + 3.56e153T^{2} \)
31 \( 1 + 3.05e78T + 3.91e156T^{2} \)
37 \( 1 - 8.35e81T + 4.58e164T^{2} \)
41 \( 1 + 4.13e84T + 2.19e169T^{2} \)
43 \( 1 - 2.16e85T + 3.26e171T^{2} \)
47 \( 1 + 2.62e87T + 3.71e175T^{2} \)
53 \( 1 + 2.71e90T + 1.11e181T^{2} \)
59 \( 1 - 1.20e93T + 8.69e185T^{2} \)
61 \( 1 - 7.31e93T + 2.88e187T^{2} \)
67 \( 1 + 2.41e95T + 5.46e191T^{2} \)
71 \( 1 + 9.07e96T + 2.41e194T^{2} \)
73 \( 1 + 9.30e97T + 4.45e195T^{2} \)
79 \( 1 - 5.92e99T + 1.78e199T^{2} \)
83 \( 1 + 1.53e99T + 3.18e201T^{2} \)
89 \( 1 - 2.87e102T + 4.85e204T^{2} \)
97 \( 1 - 8.84e103T + 4.08e208T^{2} \)
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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

−13.01319816814220653219014906374, −11.61100383043436634886484599849, −10.91824131918483754701053595767, −8.582976328233921236462955066352, −7.49703227803332663415949557961, −5.36599937372235074227998721522, −4.75050877146327859734412723270, −3.41325874474541884210822030980, −0.912040452058771340430934866568, 0, 0.912040452058771340430934866568, 3.41325874474541884210822030980, 4.75050877146327859734412723270, 5.36599937372235074227998721522, 7.49703227803332663415949557961, 8.582976328233921236462955066352, 10.91824131918483754701053595767, 11.61100383043436634886484599849, 13.01319816814220653219014906374

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