Properties

Degree 2
Conductor $ 2^{2} \cdot 7537 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 3

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 4·5-s − 4·7-s + 6·9-s − 2·11-s − 2·13-s + 12·15-s − 8·17-s − 6·19-s + 12·21-s − 5·23-s + 11·25-s − 9·27-s − 6·29-s + 3·31-s + 6·33-s + 16·35-s − 10·37-s + 6·39-s − 10·43-s − 24·45-s − 9·47-s + 9·49-s + 24·51-s − 11·53-s + 8·55-s + 18·57-s + ⋯
L(s)  = 1  − 1.73·3-s − 1.78·5-s − 1.51·7-s + 2·9-s − 0.603·11-s − 0.554·13-s + 3.09·15-s − 1.94·17-s − 1.37·19-s + 2.61·21-s − 1.04·23-s + 11/5·25-s − 1.73·27-s − 1.11·29-s + 0.538·31-s + 1.04·33-s + 2.70·35-s − 1.64·37-s + 0.960·39-s − 1.52·43-s − 3.57·45-s − 1.31·47-s + 9/7·49-s + 3.36·51-s − 1.51·53-s + 1.07·55-s + 2.38·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(30148\)    =    \(2^{2} \cdot 7537\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{30148} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(2,\ 30148,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7537\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7537\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
7537 \( 1 - T \)
good3 \( 1 + p T + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 13 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−15.97949988954495, −15.58575196513895, −15.25688452018896, −14.59611224158543, −13.36199158063591, −13.08233238789569, −12.63951074332266, −12.04606545704859, −11.81575562310738, −11.03374676236667, −10.84342070367097, −10.20855995233503, −9.680576767901322, −8.810767933335960, −8.275376657188046, −7.513866880189252, −6.937667019644035, −6.488679487289752, −6.226156822132417, −5.160997308158546, −4.725155990781252, −4.101134082349192, −3.632864472763103, −2.738409063015659, −1.713101019838388, 0, 0, 0, 1.713101019838388, 2.738409063015659, 3.632864472763103, 4.101134082349192, 4.725155990781252, 5.160997308158546, 6.226156822132417, 6.488679487289752, 6.937667019644035, 7.513866880189252, 8.275376657188046, 8.810767933335960, 9.680576767901322, 10.20855995233503, 10.84342070367097, 11.03374676236667, 11.81575562310738, 12.04606545704859, 12.63951074332266, 13.08233238789569, 13.36199158063591, 14.59611224158543, 15.25688452018896, 15.58575196513895, 15.97949988954495

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