Properties

Label 2-1050-35.27-c1-0-5
Degree $2$
Conductor $1050$
Sign $-0.372 - 0.928i$
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 + 1.46·11-s + (−0.707 + 0.707i)12-s + (0.189 + 0.189i)13-s + (1.73 + 2i)14-s − 1.00·16-s + (−3.53 + 3.53i)17-s + (0.707 − 0.707i)18-s − 0.535·19-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 + 0.441·11-s + (−0.204 + 0.204i)12-s + (0.0525 + 0.0525i)13-s + (0.462 + 0.534i)14-s − 0.250·16-s + (−0.857 + 0.857i)17-s + (0.166 − 0.166i)18-s − 0.122·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.372 - 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.372 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.372 - 0.928i$
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: \((2,\ 1050,\ (\ :1/2),\ -0.372 - 0.928i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6732315700\)
\(L(\frac12)\) \(\approx\) \(0.6732315700\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.46T + 11T^{2} \)
13 \( 1 + (-0.189 - 0.189i)T + 13iT^{2} \)
17 \( 1 + (3.53 - 3.53i)T - 17iT^{2} \)
19 \( 1 + 0.535T + 19T^{2} \)
23 \( 1 + (0.707 - 0.707i)T - 23iT^{2} \)
29 \( 1 - 4.26iT - 29T^{2} \)
31 \( 1 - 5.92iT - 31T^{2} \)
37 \( 1 + (4.89 + 4.89i)T + 37iT^{2} \)
41 \( 1 - 2.26iT - 41T^{2} \)
43 \( 1 + (0.189 - 0.189i)T - 43iT^{2} \)
47 \( 1 + (8.76 - 8.76i)T - 47iT^{2} \)
53 \( 1 + (7.39 - 7.39i)T - 53iT^{2} \)
59 \( 1 - 3.19T + 59T^{2} \)
61 \( 1 - 4.46iT - 61T^{2} \)
67 \( 1 + (10.1 + 10.1i)T + 67iT^{2} \)
71 \( 1 + 4.53T + 71T^{2} \)
73 \( 1 + (-5.93 - 5.93i)T + 73iT^{2} \)
79 \( 1 + 8.39iT - 79T^{2} \)
83 \( 1 + (4.57 + 4.57i)T + 83iT^{2} \)
89 \( 1 + 12T + 89T^{2} \)
97 \( 1 + (-11.9 + 11.9i)T - 97iT^{2} \)
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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

−10.24002919525725996450334018922, −9.155318012082608018497786623372, −8.949289181977190913279793130026, −7.87131780835003688709483626590, −6.87309997382606613579090816718, −6.09782966285837653151526583837, −4.68288591482310508917966993982, −3.71511074113471462979507680920, −2.95796146022287404728338367227, −1.64174781137156115278535678415, 0.32775217798076536249618302901, 2.02143903838823030023239542793, 3.18440704779564779444308599372, 4.36721071814325537192498019149, 5.64111622739156794672766038045, 6.59759048945552298069561997534, 6.98860228431715994847760210739, 8.072005353133617370839819848716, 8.794520426249600782979200712713, 9.569239352162119616663383137162

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