Properties

Degree $2$
Conductor $248430$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s + 5-s − 6-s − 8-s + 9-s − 10-s − 4·11-s + 12-s + 15-s + 16-s + 6·17-s − 18-s + 4·19-s + 20-s + 4·22-s + 8·23-s − 24-s + 25-s + 27-s + 6·29-s − 30-s − 8·31-s − 32-s − 4·33-s − 6·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s + 0.288·12-s + 0.258·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.852·22-s + 1.66·23-s − 0.204·24-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 0.182·30-s − 1.43·31-s − 0.176·32-s − 0.696·33-s − 1.02·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(248430\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{248430} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 248430,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.970542648\)
\(L(\frac12)\) \(\approx\) \(2.970542648\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 \)
13 \( 1 \)
good11 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−12.94143707800738, −12.55589581491306, −11.85149051829636, −11.35246471697066, −10.94186546154875, −10.35706503928258, −10.00811640296978, −9.645951260950627, −9.165644764571078, −8.663444220402711, −8.198481779275227, −7.665754135027934, −7.385279994187499, −6.933149445101411, −6.216715469947628, −5.657087493089499, −5.218283055254372, −4.819675558296153, −3.977202579743523, −3.205178889780719, −2.923019477717710, −2.530896066537913, −1.640737721334983, −1.183078341123900, −0.5422387604636218, 0.5422387604636218, 1.183078341123900, 1.640737721334983, 2.530896066537913, 2.923019477717710, 3.205178889780719, 3.977202579743523, 4.819675558296153, 5.218283055254372, 5.657087493089499, 6.216715469947628, 6.933149445101411, 7.385279994187499, 7.665754135027934, 8.198481779275227, 8.663444220402711, 9.165644764571078, 9.645951260950627, 10.00811640296978, 10.35706503928258, 10.94186546154875, 11.35246471697066, 11.85149051829636, 12.55589581491306, 12.94143707800738

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