↑

Properties

Label 2-1050-15.2-c1-0-2
Degree $2$
Conductor $1050$
Sign $-0.658 - 0.752i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)2-s + (−0.510 + 1.65i)3-s − 1.00i·4-s + (0.809 + 1.53i)6-s + (−0.707 − 0.707i)7-s + (−0.707 − 0.707i)8-s + (−2.47 − 1.68i)9-s − 0.598i·11-s + (1.65 + 0.510i)12-s + (−2.55 + 2.55i)13-s − 1.00·14-s − 1.00·16-s + (−4.20 + 4.20i)17-s + (−2.94 + 0.558i)18-s + 5.70i·19-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.294 + 0.955i)3-s − 0.500i·4-s + (0.330 + 0.625i)6-s + (−0.267 − 0.267i)7-s + (−0.250 − 0.250i)8-s + (−0.826 − 0.563i)9-s − 0.180i·11-s + (0.477 + 0.147i)12-s + (−0.709 + 0.709i)13-s − 0.267·14-s − 0.250·16-s + (−1.01 + 1.01i)17-s + (−0.694 + 0.131i)18-s + 1.30i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.658 - 0.752i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ -0.658 - 0.752i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7184186453\)
\(L(\frac12)\) \(\approx\) \(0.7184186453\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.510 - 1.65i)T \)
5 \( 1 \)
7 \( 1 + (0.707 + 0.707i)T \)
good11 \( 1 + 0.598iT - 11T^{2} \)
13 \( 1 + (2.55 - 2.55i)T - 13iT^{2} \)
17 \( 1 + (4.20 - 4.20i)T - 17iT^{2} \)
19 \( 1 - 5.70iT - 19T^{2} \)
23 \( 1 + (-2.23 - 2.23i)T + 23iT^{2} \)
29 \( 1 + 0.0410T + 29T^{2} \)
31 \( 1 + 8.68T + 31T^{2} \)
37 \( 1 + (1.56 + 1.56i)T + 37iT^{2} \)
41 \( 1 - 5.79iT - 41T^{2} \)
43 \( 1 + (0.325 - 0.325i)T - 43iT^{2} \)
47 \( 1 + (-1.56 + 1.56i)T - 47iT^{2} \)
53 \( 1 + (-2.01 - 2.01i)T + 53iT^{2} \)
59 \( 1 + 9.35T + 59T^{2} \)
61 \( 1 + 14.8T + 61T^{2} \)
67 \( 1 + (5.89 + 5.89i)T + 67iT^{2} \)
71 \( 1 - 14.4iT - 71T^{2} \)
73 \( 1 + (-9.67 + 9.67i)T - 73iT^{2} \)
79 \( 1 + 11.7iT - 79T^{2} \)
83 \( 1 + (-1.04 - 1.04i)T + 83iT^{2} \)
89 \( 1 - 18.1T + 89T^{2} \)
97 \( 1 + (-4.69 - 4.69i)T + 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.49367418877306555853858705149, −9.446031722375401201858706143294, −8.999283952111630213723775627146, −7.73411898373767273773372573612, −6.52419391950858996717131293912, −5.81431713884985646640942462126, −4.81085549391503754343241418970, −4.02840046732028538751754205751, −3.27041899434247896163102271885, −1.84333385906558930820971196978, 0.26291409854503534232253361889, 2.25127563947368065169046050280, 3.07408462412229362548018653084, 4.70078822945825778477334311249, 5.31707975391653175489919184774, 6.31906194144407292056632661823, 7.11533186610785225869475159884, 7.53575758495078532793994173137, 8.729932313757725449981922900687, 9.300619218669888133142274584497

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