Properties

Degree $2$
Conductor $1050$
Sign $0.487 - 0.872i$
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 + (1.61 + 0.618i)3-s − 4-s + (0.618 − 1.61i)6-s + (−2.61 − 0.381i)7-s + i·8-s + (2.23 + 2.00i)9-s + 4.47i·11-s + (−1.61 − 0.618i)12-s + 1.23i·13-s + (−0.381 + 2.61i)14-s + 16-s − 5.23·17-s + (2.00 − 2.23i)18-s + 8.47i·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.934 + 0.356i)3-s − 0.5·4-s + (0.252 − 0.660i)6-s + (−0.989 − 0.144i)7-s + 0.353i·8-s + (0.745 + 0.666i)9-s + 1.34i·11-s + (−0.467 − 0.178i)12-s + 0.342i·13-s + (−0.102 + 0.699i)14-s + 0.250·16-s − 1.26·17-s + (0.471 − 0.527i)18-s + 1.94i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.487 - 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.487 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.487 - 0.872i$
Motivic weight: \(1\)
Character: $\chi_{1050} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ 0.487 - 0.872i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.485239903\)
\(L(\frac12)\) \(\approx\) \(1.485239903\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (-1.61 - 0.618i)T \)
5 \( 1 \)
7 \( 1 + (2.61 + 0.381i)T \)
good11 \( 1 - 4.47iT - 11T^{2} \)
13 \( 1 - 1.23iT - 13T^{2} \)
17 \( 1 + 5.23T + 17T^{2} \)
19 \( 1 - 8.47iT - 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 + 7.70iT - 29T^{2} \)
31 \( 1 + 2.76iT - 31T^{2} \)
37 \( 1 + 0.763T + 37T^{2} \)
41 \( 1 + 2.47T + 41T^{2} \)
43 \( 1 - 4.94T + 43T^{2} \)
47 \( 1 - 6.47T + 47T^{2} \)
53 \( 1 + 0.472iT - 53T^{2} \)
59 \( 1 - 4.47T + 59T^{2} \)
61 \( 1 - 7.23iT - 61T^{2} \)
67 \( 1 - 12T + 67T^{2} \)
71 \( 1 + 7.23iT - 71T^{2} \)
73 \( 1 - 11.2iT - 73T^{2} \)
79 \( 1 + 8.94T + 79T^{2} \)
83 \( 1 + 14.6T + 83T^{2} \)
89 \( 1 + 5.52T + 89T^{2} \)
97 \( 1 + 0.763iT - 97T^{2} \)
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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

−9.805372156430722560620490400632, −9.598556776476758320000296098527, −8.617202945280402515977981238404, −7.69117245278843988481040447618, −6.87171882124857471827774169151, −5.66133238956742418883971715771, −4.20956438804436174170426597315, −3.95604944487530147394437667590, −2.63752327217385364092547780433, −1.80268033401864132652597302012, 0.58126049432332938940186507270, 2.59889782380549954575825303729, 3.33776836665511901497326538238, 4.48036056640841011429084416583, 5.68556478748470897119337239977, 6.81783940051827124334884273252, 6.91967579864884439934010747571, 8.328535453250362704579344269396, 8.839834757276002089325093229443, 9.300508467250497076700932491945

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