Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.993 + 0.116i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (2.16 + 1.52i)7-s + 0.999·8-s + (4.29 + 2.48i)11-s + 5.49·13-s + (0.243 − 2.63i)14-s + (−0.5 − 0.866i)16-s + (−2.66 − 1.53i)17-s + (2.68 − 1.55i)19-s − 4.96i·22-s + (3.08 + 5.34i)23-s + (−2.74 − 4.75i)26-s + (−2.40 + 1.10i)28-s − 6.67i·29-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.816 + 0.577i)7-s + 0.353·8-s + (1.29 + 0.748i)11-s + 1.52·13-s + (0.0649 − 0.704i)14-s + (−0.125 − 0.216i)16-s + (−0.645 − 0.372i)17-s + (0.616 − 0.355i)19-s − 1.05i·22-s + (0.643 + 1.11i)23-s + (−0.538 − 0.933i)26-s + (−0.454 + 0.209i)28-s − 1.24i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.116i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.993 + 0.116i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (899, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.993 + 0.116i)$
$L(1)$  $\approx$  $2.032706688$
$L(\frac12)$  $\approx$  $2.032706688$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (-2.16 - 1.52i)T \)
good11 \( 1 + (-4.29 - 2.48i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.49T + 13T^{2} \)
17 \( 1 + (2.66 + 1.53i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.68 + 1.55i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.08 - 5.34i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.67iT - 29T^{2} \)
31 \( 1 + (1.01 + 0.586i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-9.27 + 5.35i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.39T + 41T^{2} \)
43 \( 1 - 8.81iT - 43T^{2} \)
47 \( 1 + (3.59 - 2.07i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.22 + 3.85i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.00 - 5.20i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9.05 + 5.22i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.3 + 5.97i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.973iT - 71T^{2} \)
73 \( 1 + (8.34 - 14.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.12 + 3.67i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 14.2iT - 83T^{2} \)
89 \( 1 + (7.38 + 12.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 4.41T + 97T^{2} \)
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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

−8.845668588872647426750828035796, −8.091296825745834823167439983424, −7.31277797075660528014061709972, −6.43676321824019025800277171104, −5.61009022444790218570691063384, −4.57793631365576700489415027496, −3.96497969760451623072953162992, −2.93980247930746861182821653312, −1.82737092738703946143744489086, −1.12446845233006850054230006457, 0.936949797295644206587327979160, 1.60922967449158044086794047638, 3.31852843090361341534482983541, 4.06531791424866426746025570430, 4.87990317337946477754891664796, 5.87425185226814943266819597310, 6.51774050708750867025214339534, 7.10774019275121141974724758274, 8.102120508318499091920321864543, 8.755111021141802138713225288473

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