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Properties

Label 2-1050-21.20-c1-0-26
Degree $2$
Conductor $1050$
Sign $0.978 - 0.208i$
Analytic cond. $8.38429$
Root an. cond. $2.89556$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s + (1.68 + 0.420i)3-s − 4-s + (0.420 − 1.68i)6-s + (1.16 + 2.37i)7-s + i·8-s + (2.64 + 1.41i)9-s + 2.82i·11-s + (−1.68 − 0.420i)12-s − 0.841i·13-s + (2.37 − 1.16i)14-s + 16-s − 1.19·17-s + (1.41 − 2.64i)18-s − 4.55i·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.970 + 0.242i)3-s − 0.5·4-s + (0.171 − 0.685i)6-s + (0.439 + 0.898i)7-s + 0.353i·8-s + (0.881 + 0.471i)9-s + 0.852i·11-s + (−0.485 − 0.121i)12-s − 0.233i·13-s + (0.635 − 0.311i)14-s + 0.250·16-s − 0.288·17-s + (0.333 − 0.623i)18-s − 1.04i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.208i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 - 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.978 - 0.208i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ 0.978 - 0.208i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.228303088\)
\(L(\frac12)\) \(\approx\) \(2.228303088\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.68 - 0.420i)T \)
5 \( 1 \)
7 \( 1 + (-1.16 - 2.37i)T \)
good11 \( 1 - 2.82iT - 11T^{2} \)
13 \( 1 + 0.841iT - 13T^{2} \)
17 \( 1 + 1.19T + 17T^{2} \)
19 \( 1 + 4.55iT - 19T^{2} \)
23 \( 1 - 3.29iT - 23T^{2} \)
29 \( 1 - 7.98iT - 29T^{2} \)
31 \( 1 - 5.53iT - 31T^{2} \)
37 \( 1 - 10.8T + 37T^{2} \)
41 \( 1 + 7.82T + 41T^{2} \)
43 \( 1 - 4.65T + 43T^{2} \)
47 \( 1 - 4.33T + 47T^{2} \)
53 \( 1 + 12.5iT - 53T^{2} \)
59 \( 1 + 3.91T + 59T^{2} \)
61 \( 1 + 10.0iT - 61T^{2} \)
67 \( 1 - 4.65T + 67T^{2} \)
71 \( 1 + 12.6iT - 71T^{2} \)
73 \( 1 - 3.06iT - 73T^{2} \)
79 \( 1 - 7.29T + 79T^{2} \)
83 \( 1 - 7.70T + 83T^{2} \)
89 \( 1 + 12.8T + 89T^{2} \)
97 \( 1 + 8.11iT - 97T^{2} \)
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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

−9.765595169048570152343452642227, −9.192793716847264568032830745186, −8.542410998543180802779877461929, −7.69833020680841368745783410173, −6.73571852957405642929390376455, −5.20627364600485544027003704801, −4.65440299530035597327839077276, −3.43540531765408701190554538034, −2.53903713272494110986143488104, −1.63594264162412229358345708799, 0.989153278428979362414854182035, 2.51792064107248490324160511821, 3.89524228511810608492648182611, 4.36584231390007882062990898359, 5.84425615648474443592169376406, 6.60928982506391706704919941541, 7.68236822228043597606354948366, 7.987138579391957506962120468426, 8.853186546869516613146228222664, 9.706963130696336282715639793136

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