Properties

Degree $2$
Conductor $1050$
Sign $0.674 - 0.738i$
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 + (−1.68 − 0.403i)3-s − 4-s + (−0.403 + 1.68i)6-s + (−2.31 − 1.28i)7-s + i·8-s + (2.67 + 1.35i)9-s − 5.34i·11-s + (1.68 + 0.403i)12-s + 3.95i·13-s + (−1.28 + 2.31i)14-s + 16-s − 7.32·17-s + (1.35 − 2.67i)18-s − 0.807i·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.972 − 0.232i)3-s − 0.5·4-s + (−0.164 + 0.687i)6-s + (−0.874 − 0.484i)7-s + 0.353i·8-s + (0.891 + 0.453i)9-s − 1.61i·11-s + (0.486 + 0.116i)12-s + 1.09i·13-s + (−0.342 + 0.618i)14-s + 0.250·16-s − 1.77·17-s + (0.320 − 0.630i)18-s − 0.185i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.674 - 0.738i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.674 - 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.674 - 0.738i$
Motivic weight: \(1\)
Character: $\chi_{1050} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ 0.674 - 0.738i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3411451917\)
\(L(\frac12)\) \(\approx\) \(0.3411451917\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.68 + 0.403i)T \)
5 \( 1 \)
7 \( 1 + (2.31 + 1.28i)T \)
good11 \( 1 + 5.34iT - 11T^{2} \)
13 \( 1 - 3.95iT - 13T^{2} \)
17 \( 1 + 7.32T + 17T^{2} \)
19 \( 1 + 0.807iT - 19T^{2} \)
23 \( 1 - 0.281iT - 23T^{2} \)
29 \( 1 - 0.281iT - 29T^{2} \)
31 \( 1 - 9.07iT - 31T^{2} \)
37 \( 1 - 6.06T + 37T^{2} \)
41 \( 1 + 6.15T + 41T^{2} \)
43 \( 1 - 6.34T + 43T^{2} \)
47 \( 1 - 5.78T + 47T^{2} \)
53 \( 1 - 10.9iT - 53T^{2} \)
59 \( 1 - 4.90T + 59T^{2} \)
61 \( 1 - 13.2iT - 61T^{2} \)
67 \( 1 + 6.71T + 67T^{2} \)
71 \( 1 - 3.36iT - 71T^{2} \)
73 \( 1 + 4.98iT - 73T^{2} \)
79 \( 1 + 3.26T + 79T^{2} \)
83 \( 1 - 1.53T + 83T^{2} \)
89 \( 1 - 4.31T + 89T^{2} \)
97 \( 1 + 15.0iT - 97T^{2} \)
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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

−10.39175290015866573581120027654, −9.197025479264362243770763892984, −8.719798572341845104316181932844, −7.29776420317175824181057322759, −6.52473388509827624283045884607, −5.86593486433773230876238051548, −4.63461774457505262173126808118, −3.86742693588533508918653570831, −2.61841997007074116825203943452, −1.10115525474682899880370079157, 0.20222678518793418210958281516, 2.29685301149355771755722748684, 3.92685046050120773943335277086, 4.73292835751712898114803004619, 5.62625544543816966118978708191, 6.42555952565073363198174032958, 7.03090360719705128488188985460, 7.964834957060348081954651790464, 9.191700137400871395004750432280, 9.734139839663403187832256306593

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