Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5^{2} \cdot 7 $
Sign $-0.218 - 0.975i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·2-s + (0.618 + 1.61i)3-s − 4-s + (1.61 − 0.618i)6-s + (−0.381 + 2.61i)7-s + i·8-s + (−2.23 + 2.00i)9-s − 4.47i·11-s + (−0.618 − 1.61i)12-s + 3.23i·13-s + (2.61 + 0.381i)14-s + 16-s + 0.763·17-s + (2.00 + 2.23i)18-s + 0.472i·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.356 + 0.934i)3-s − 0.5·4-s + (0.660 − 0.252i)6-s + (−0.144 + 0.989i)7-s + 0.353i·8-s + (−0.745 + 0.666i)9-s − 1.34i·11-s + (−0.178 − 0.467i)12-s + 0.897i·13-s + (0.699 + 0.102i)14-s + 0.250·16-s + 0.185·17-s + (0.471 + 0.527i)18-s + 0.108i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.218 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.218 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.218 - 0.975i$
motivic weight  =  \(1\)
character  :  $\chi_{1050} (251, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1050,\ (\ :1/2),\ -0.218 - 0.975i)\)
\(L(1)\)  \(\approx\)  \(1.167089200\)
\(L(\frac12)\)  \(\approx\)  \(1.167089200\)
\(L(\frac{3}{2})\)   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 \( 1 + iT \)
3 \( 1 + (-0.618 - 1.61i)T \)
5 \( 1 \)
7 \( 1 + (0.381 - 2.61i)T \)
good11 \( 1 + 4.47iT - 11T^{2} \)
13 \( 1 - 3.23iT - 13T^{2} \)
17 \( 1 - 0.763T + 17T^{2} \)
19 \( 1 - 0.472iT - 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 - 5.70iT - 29T^{2} \)
31 \( 1 - 7.23iT - 31T^{2} \)
37 \( 1 + 5.23T + 37T^{2} \)
41 \( 1 + 6.47T + 41T^{2} \)
43 \( 1 + 12.9T + 43T^{2} \)
47 \( 1 - 2.47T + 47T^{2} \)
53 \( 1 - 8.47iT - 53T^{2} \)
59 \( 1 - 4.47T + 59T^{2} \)
61 \( 1 + 2.76iT - 61T^{2} \)
67 \( 1 - 12T + 67T^{2} \)
71 \( 1 + 2.76iT - 71T^{2} \)
73 \( 1 + 6.76iT - 73T^{2} \)
79 \( 1 - 8.94T + 79T^{2} \)
83 \( 1 + 16.6T + 83T^{2} \)
89 \( 1 - 14.4T + 89T^{2} \)
97 \( 1 - 5.23iT - 97T^{2} \)
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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

−10.17304368669509697110525662727, −9.248471301113782674708941166278, −8.787612149401755600018459855155, −8.167227642151999366900971417907, −6.65203393111786112308311253682, −5.51249551236562110079772572073, −4.95237862709696937451527161312, −3.57024867946714134826244869177, −3.12933767218580079014471923568, −1.80681704106805753861817167884, 0.49127357336985272608076892653, 2.02127906889695795207952231546, 3.40679157353147941679105701482, 4.46366704691807538583500323137, 5.54087752498860918235353038522, 6.67117526894759038248053968955, 7.07695037548084124573798084643, 7.934332356573024716128761210663, 8.450759732580279291586097606746, 9.769462170824812422124966789639

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