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Properties

Label 2-1050-35.4-c1-0-23
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  + (0.866 − 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s − 0.999·6-s + (−0.866 − 2.5i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (−1.5 + 2.59i)11-s + (−0.866 + 0.499i)12-s − 4i·13-s + (−2 − 1.73i)14-s + (−0.5 − 0.866i)16-s + (0.866 + 0.499i)18-s + (−2 − 3.46i)19-s + (−0.500 + 2.59i)21-s + 3i·22-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.499 − 0.288i)3-s + (0.249 − 0.433i)4-s − 0.408·6-s + (−0.327 − 0.944i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.452 + 0.783i)11-s + (−0.249 + 0.144i)12-s − 1.10i·13-s + (−0.534 − 0.462i)14-s + (−0.125 − 0.216i)16-s + (0.204 + 0.117i)18-s + (−0.458 − 0.794i)19-s + (−0.109 + 0.566i)21-s + 0.639i·22-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: $\chi_{1050} (949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ -0.978 + 0.208i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.119959802\)
\(L(\frac12)\) \(\approx\) \(1.119959802\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 \)
7 \( 1 + (0.866 + 2.5i)T \)
good11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 9T + 29T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.92 - 4i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 10iT - 43T^{2} \)
47 \( 1 + (-5.19 + 3i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.59 - 1.5i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.66 + 5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (1.73 + i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9iT - 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - iT - 97T^{2} \)
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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

−9.951922763569168722933775795772, −8.734675974101682933821695142241, −7.42456191576172860577766348488, −7.11981284068111357386184150054, −5.95925553933719211777171711589, −5.16026073419900340650664632823, −4.25596084819465612798249005972, −3.22631406184955385514933681809, −1.95339290357509476620194273383, −0.41142353217782761279112619525, 2.01929321309515344089331295261, 3.29918213439939969480557388969, 4.22758350827431420068496720724, 5.33658826514523693202033472041, 5.90794708561468377535199588299, 6.65203283044843767897352233788, 7.70375136268900109409239340842, 8.718316171634410754785715517045, 9.341163992801231924837272116470, 10.41061984758202901319370610274

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