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Properties

Label 2-1050-35.12-c1-0-19
Degree $2$
Conductor $1050$
Sign $-0.109 + 0.993i$
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  + (0.965 − 0.258i)2-s + (−0.258 + 0.965i)3-s + (0.866 − 0.499i)4-s + i·6-s + (−1.42 − 2.22i)7-s + (0.707 − 0.707i)8-s + (−0.866 − 0.499i)9-s + (−0.230 − 0.399i)11-s + (0.258 + 0.965i)12-s + (−4.00 − 4.00i)13-s + (−1.95 − 1.78i)14-s + (0.500 − 0.866i)16-s + (−1.58 − 0.424i)17-s + (−0.965 − 0.258i)18-s + (2.91 − 5.04i)19-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (−0.149 + 0.557i)3-s + (0.433 − 0.249i)4-s + 0.408i·6-s + (−0.539 − 0.841i)7-s + (0.249 − 0.249i)8-s + (−0.288 − 0.166i)9-s + (−0.0695 − 0.120i)11-s + (0.0747 + 0.278i)12-s + (−1.11 − 1.11i)13-s + (−0.522 − 0.476i)14-s + (0.125 − 0.216i)16-s + (−0.384 − 0.102i)17-s + (−0.227 − 0.0610i)18-s + (0.668 − 1.15i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.109 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.109 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.109 + 0.993i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ -0.109 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.592064079\)
\(L(\frac12)\) \(\approx\) \(1.592064079\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
3 \( 1 + (0.258 - 0.965i)T \)
5 \( 1 \)
7 \( 1 + (1.42 + 2.22i)T \)
good11 \( 1 + (0.230 + 0.399i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.00 + 4.00i)T + 13iT^{2} \)
17 \( 1 + (1.58 + 0.424i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-2.91 + 5.04i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.14 + 4.26i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 5.53iT - 29T^{2} \)
31 \( 1 + (-0.0280 + 0.0162i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.78 + 2.08i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 10.9iT - 41T^{2} \)
43 \( 1 + (4.75 - 4.75i)T - 43iT^{2} \)
47 \( 1 + (-2.33 - 8.73i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.65 - 0.710i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.958 + 1.66i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (11.7 + 6.77i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.986 + 3.68i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 8.85T + 71T^{2} \)
73 \( 1 + (-1.05 + 3.91i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-4.38 - 2.53i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.08 - 1.08i)T + 83iT^{2} \)
89 \( 1 + (-5.71 + 9.89i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.51 + 2.51i)T - 97iT^{2} \)
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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

−9.832968547901956855156993910857, −9.157168708318873091533095337171, −7.82815840907753594733076664431, −7.08281360890254106747565706824, −6.18072624850301872364700346323, −5.11326759158996147639925497499, −4.51680954204277372983896323447, −3.38666212052689087941787780438, −2.60970734777057501295812349546, −0.55186434480874450466609073058, 1.85060340321478397977196481582, 2.77434510828859470833635273683, 3.99746197215124697615483077368, 5.08598939640395740770659141978, 5.92241244597687757196475293402, 6.62361980058305939064189194253, 7.48183489317656574278337093480, 8.269730832028858846687299777309, 9.430966081079562790664094795080, 9.957788923696074646210663551190

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