Properties

Degree 2
Conductor $ 2^{2} \cdot 19 \cdot 53 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.17·3-s − 4.27·5-s − 4.73·7-s + 7.07·9-s + 3.63·11-s + 3.74·13-s − 13.5·15-s − 1.80·17-s − 19-s − 15.0·21-s − 2.48·23-s + 13.2·25-s + 12.9·27-s + 0.417·29-s − 6.42·31-s + 11.5·33-s + 20.2·35-s − 4.97·37-s + 11.8·39-s − 0.0129·41-s − 11.1·43-s − 30.2·45-s − 6.11·47-s + 15.4·49-s − 5.72·51-s + 53-s − 15.5·55-s + ⋯
L(s)  = 1  + 1.83·3-s − 1.91·5-s − 1.78·7-s + 2.35·9-s + 1.09·11-s + 1.03·13-s − 3.50·15-s − 0.437·17-s − 0.229·19-s − 3.28·21-s − 0.518·23-s + 2.65·25-s + 2.49·27-s + 0.0774·29-s − 1.15·31-s + 2.00·33-s + 3.42·35-s − 0.818·37-s + 1.90·39-s − 0.00202·41-s − 1.70·43-s − 4.51·45-s − 0.892·47-s + 2.20·49-s − 0.802·51-s + 0.137·53-s − 2.09·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4028\)    =    \(2^{2} \cdot 19 \cdot 53\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4028} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 4028,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;19,\;53\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;19,\;53\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
19 \( 1 + T \)
53 \( 1 - T \)
good3 \( 1 - 3.17T + 3T^{2} \)
5 \( 1 + 4.27T + 5T^{2} \)
7 \( 1 + 4.73T + 7T^{2} \)
11 \( 1 - 3.63T + 11T^{2} \)
13 \( 1 - 3.74T + 13T^{2} \)
17 \( 1 + 1.80T + 17T^{2} \)
23 \( 1 + 2.48T + 23T^{2} \)
29 \( 1 - 0.417T + 29T^{2} \)
31 \( 1 + 6.42T + 31T^{2} \)
37 \( 1 + 4.97T + 37T^{2} \)
41 \( 1 + 0.0129T + 41T^{2} \)
43 \( 1 + 11.1T + 43T^{2} \)
47 \( 1 + 6.11T + 47T^{2} \)
59 \( 1 + 12.4T + 59T^{2} \)
61 \( 1 - 4.40T + 61T^{2} \)
67 \( 1 + 14.2T + 67T^{2} \)
71 \( 1 - 4.24T + 71T^{2} \)
73 \( 1 + 10.7T + 73T^{2} \)
79 \( 1 + 3.93T + 79T^{2} \)
83 \( 1 + 11.0T + 83T^{2} \)
89 \( 1 - 3.91T + 89T^{2} \)
97 \( 1 - 7.08T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.257788940175544047310464952898, −7.38552275071253152175348585419, −6.90469739539470310863966432608, −6.25824946631262215487455060890, −4.49834066949631912861439950509, −3.79910321016174721889732950299, −3.49394881427489071240017062017, −2.96872277978238233703950482808, −1.54097321056449223719233099180, 0, 1.54097321056449223719233099180, 2.96872277978238233703950482808, 3.49394881427489071240017062017, 3.79910321016174721889732950299, 4.49834066949631912861439950509, 6.25824946631262215487455060890, 6.90469739539470310863966432608, 7.38552275071253152175348585419, 8.257788940175544047310464952898

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