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Properties

Label 2-1050-5.3-c2-0-11
Degree $2$
Conductor $1050$
Sign $0.973 + 0.229i$
Analytic cond. $28.6104$
Root an. cond. $5.34887$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − i)2-s + (−1.22 − 1.22i)3-s − 2i·4-s − 2.44·6-s + (−1.87 + 1.87i)7-s + (−2 − 2i)8-s + 2.99i·9-s − 14.7·11-s + (−2.44 + 2.44i)12-s + (13.2 + 13.2i)13-s + 3.74i·14-s − 4·16-s + (−1.17 + 1.17i)17-s + (2.99 + 2.99i)18-s − 15.1i·19-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (−0.408 − 0.408i)3-s − 0.5i·4-s − 0.408·6-s + (−0.267 + 0.267i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s − 1.34·11-s + (−0.204 + 0.204i)12-s + (1.02 + 1.02i)13-s + 0.267i·14-s − 0.250·16-s + (−0.0693 + 0.0693i)17-s + (0.166 + 0.166i)18-s − 0.796i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.973 + 0.229i$
Analytic conductor: \(28.6104\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1),\ 0.973 + 0.229i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.809819166\)
\(L(\frac12)\) \(\approx\) \(1.809819166\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
3 \( 1 + (1.22 + 1.22i)T \)
5 \( 1 \)
7 \( 1 + (1.87 - 1.87i)T \)
good11 \( 1 + 14.7T + 121T^{2} \)
13 \( 1 + (-13.2 - 13.2i)T + 169iT^{2} \)
17 \( 1 + (1.17 - 1.17i)T - 289iT^{2} \)
19 \( 1 + 15.1iT - 361T^{2} \)
23 \( 1 + (-22.5 - 22.5i)T + 529iT^{2} \)
29 \( 1 - 3.29iT - 841T^{2} \)
31 \( 1 - 50.0T + 961T^{2} \)
37 \( 1 + (6.15 - 6.15i)T - 1.36e3iT^{2} \)
41 \( 1 + 35.1T + 1.68e3T^{2} \)
43 \( 1 + (-58.9 - 58.9i)T + 1.84e3iT^{2} \)
47 \( 1 + (-20.1 + 20.1i)T - 2.20e3iT^{2} \)
53 \( 1 + (10.9 + 10.9i)T + 2.80e3iT^{2} \)
59 \( 1 - 66.6iT - 3.48e3T^{2} \)
61 \( 1 + 4.45T + 3.72e3T^{2} \)
67 \( 1 + (-88.4 + 88.4i)T - 4.48e3iT^{2} \)
71 \( 1 - 69.3T + 5.04e3T^{2} \)
73 \( 1 + (58.4 + 58.4i)T + 5.32e3iT^{2} \)
79 \( 1 - 52.8iT - 6.24e3T^{2} \)
83 \( 1 + (-53.4 - 53.4i)T + 6.88e3iT^{2} \)
89 \( 1 - 21.3iT - 7.92e3T^{2} \)
97 \( 1 + (90.3 - 90.3i)T - 9.40e3iT^{2} \)
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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

−9.805609776106109774455738218577, −8.963533509428378753160279640902, −8.019283426965159426960549302641, −6.95346181624667963208560796390, −6.21969888186251953000533419999, −5.30977599994265393627793361772, −4.52899561093156201154399606727, −3.24807377796628518007152322276, −2.29712584323578554284204812146, −0.995209783299935994784338363496, 0.62287949551522958202647526280, 2.68369034657549167142695435958, 3.60302812908608656312067628263, 4.65593469257587021897538937392, 5.49144683902183767689569167637, 6.15597645624170669021702042225, 7.13133983198340051618404590271, 8.093437752082604910027944725958, 8.673755850789258528979196293675, 9.994453708395181059525572337717

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