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Properties

Label 2-1050-35.27-c1-0-6
Degree $2$
Conductor $1050$
Sign $0.835 - 0.550i$
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  + (−0.707 − 0.707i)2-s + (−0.707 − 0.707i)3-s + 1.00i·4-s + 1.00i·6-s + (−2.63 − 0.189i)7-s + (0.707 − 0.707i)8-s + 1.00i·9-s − 5.46·11-s + (0.707 − 0.707i)12-s + (−2.63 − 2.63i)13-s + (1.73 + 2i)14-s − 1.00·16-s + (3.53 − 3.53i)17-s + (0.707 − 0.707i)18-s + 7.46·19-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.408 − 0.408i)3-s + 0.500i·4-s + 0.408i·6-s + (−0.997 − 0.0716i)7-s + (0.250 − 0.250i)8-s + 0.333i·9-s − 1.64·11-s + (0.204 − 0.204i)12-s + (−0.731 − 0.731i)13-s + (0.462 + 0.534i)14-s − 0.250·16-s + (0.857 − 0.857i)17-s + (0.166 − 0.166i)18-s + 1.71·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 - 0.550i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.835 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.835 - 0.550i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ 0.835 - 0.550i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5061069727\)
\(L(\frac12)\) \(\approx\) \(0.5061069727\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 \)
7 \( 1 + (2.63 + 0.189i)T \)
good11 \( 1 + 5.46T + 11T^{2} \)
13 \( 1 + (2.63 + 2.63i)T + 13iT^{2} \)
17 \( 1 + (-3.53 + 3.53i)T - 17iT^{2} \)
19 \( 1 - 7.46T + 19T^{2} \)
23 \( 1 + (0.707 - 0.707i)T - 23iT^{2} \)
29 \( 1 - 7.73iT - 29T^{2} \)
31 \( 1 - 7.92iT - 31T^{2} \)
37 \( 1 + (-4.89 - 4.89i)T + 37iT^{2} \)
41 \( 1 + 5.73iT - 41T^{2} \)
43 \( 1 + (2.63 - 2.63i)T - 43iT^{2} \)
47 \( 1 + (5.93 - 5.93i)T - 47iT^{2} \)
53 \( 1 + (2.50 - 2.50i)T - 53iT^{2} \)
59 \( 1 - 7.19T + 59T^{2} \)
61 \( 1 - 2.46iT - 61T^{2} \)
67 \( 1 + (-4.52 - 4.52i)T + 67iT^{2} \)
71 \( 1 + 11.4T + 71T^{2} \)
73 \( 1 + (-8.76 - 8.76i)T + 73iT^{2} \)
79 \( 1 - 12.3iT - 79T^{2} \)
83 \( 1 + (0.328 + 0.328i)T + 83iT^{2} \)
89 \( 1 - 12T + 89T^{2} \)
97 \( 1 + (2.17 - 2.17i)T - 97iT^{2} \)
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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

−10.02808423727980263066753988553, −9.473518819298288516513876663124, −8.228130969085373098020690517133, −7.48443209259661662543312096770, −6.94947736118300372501009274461, −5.51242470554794740349154046772, −5.06427181943368739568436278504, −3.17906939707581314821619776730, −2.80244749327818847353757221292, −0.998771933116725991395362808087, 0.34774485663229778599519760073, 2.37299797815533774382579524413, 3.58344312194057918639929999463, 4.86797740851374718303674989313, 5.67386903116050716916098584547, 6.35369200122807815080048936760, 7.52884445959699815268904625549, 7.943266168888798258810873926424, 9.261021964997748912659569959809, 9.919215213648432742300562831277

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