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Properties

Label 2-1050-7.2-c1-0-1
Degree $2$
Conductor $1050$
Sign $-0.991 - 0.126i$
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  + (−0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s − 0.999·6-s + (−2 − 1.73i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (2.5 + 4.33i)11-s + (0.499 − 0.866i)12-s − 13-s + (2.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (−1 − 1.73i)17-s + (−0.499 − 0.866i)18-s + (−3.5 + 6.06i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s − 0.408·6-s + (−0.755 − 0.654i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.753 + 1.30i)11-s + (0.144 − 0.249i)12-s − 0.277·13-s + (0.668 − 0.231i)14-s + (−0.125 + 0.216i)16-s + (−0.242 − 0.420i)17-s + (−0.117 − 0.204i)18-s + (−0.802 + 1.39i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.991 - 0.126i$
Analytic conductor: \(8.38429\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1/2),\ -0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7694798007\)
\(L(\frac12)\) \(\approx\) \(0.7694798007\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 \)
7 \( 1 + (2 + 1.73i)T \)
good11 \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.5 - 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (-3 - 5.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.5 - 4.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 9T + 41T^{2} \)
43 \( 1 + 10T + 43T^{2} \)
47 \( 1 + (6.5 - 11.2i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3 - 5.19i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + (-2 - 3.46i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7 + 12.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 10T + 83T^{2} \)
89 \( 1 + (5 - 8.66i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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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

−9.991161951984837507318066731912, −9.645210157765082611373812673617, −8.660917275195143204732878418658, −7.86556028155846650760199341258, −6.80126293297243804989106169909, −6.47569102092125468423382595864, −5.00275975643616090818386209874, −4.30383767725114563157243241227, −3.26473066150049806582827933688, −1.67953888743768526494966359844, 0.36677781587345830505252463513, 1.94791095758892736252609272369, 3.00034619207011288100683687900, 3.77402736942792387047970570486, 5.20223858816623040256689346701, 6.35655816207800325138944002088, 6.88595987967629537781222985573, 8.219188711250654176940103147551, 8.764660306417507112711453995194, 9.358687321404272406710989310262

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