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Properties

Label 2-1050-5.2-c2-0-10
Degree $2$
Conductor $1050$
Sign $-0.945 - 0.326i$
Analytic cond. $28.6104$
Root an. cond. $5.34887$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + i)2-s + (−1.22 + 1.22i)3-s + 2i·4-s − 2.44·6-s + (−1.87 − 1.87i)7-s + (−2 + 2i)8-s − 2.99i·9-s + 2.50·11-s + (−2.44 − 2.44i)12-s + (9.25 − 9.25i)13-s − 3.74i·14-s − 4·16-s + (10.6 + 10.6i)17-s + (2.99 − 2.99i)18-s + 28.8i·19-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (−0.408 + 0.408i)3-s + 0.5i·4-s − 0.408·6-s + (−0.267 − 0.267i)7-s + (−0.250 + 0.250i)8-s − 0.333i·9-s + 0.227·11-s + (−0.204 − 0.204i)12-s + (0.711 − 0.711i)13-s − 0.267i·14-s − 0.250·16-s + (0.624 + 0.624i)17-s + (0.166 − 0.166i)18-s + 1.51i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 - 0.326i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.945 - 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.945 - 0.326i$
Analytic conductor: \(28.6104\)
Root analytic conductor: \(5.34887\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1050} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1050,\ (\ :1),\ -0.945 - 0.326i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.468208314\)
\(L(\frac12)\) \(\approx\) \(1.468208314\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
3 \( 1 + (1.22 - 1.22i)T \)
5 \( 1 \)
7 \( 1 + (1.87 + 1.87i)T \)
good11 \( 1 - 2.50T + 121T^{2} \)
13 \( 1 + (-9.25 + 9.25i)T - 169iT^{2} \)
17 \( 1 + (-10.6 - 10.6i)T + 289iT^{2} \)
19 \( 1 - 28.8iT - 361T^{2} \)
23 \( 1 + (8.08 - 8.08i)T - 529iT^{2} \)
29 \( 1 - 34.8iT - 841T^{2} \)
31 \( 1 + 37.6T + 961T^{2} \)
37 \( 1 + (-12.2 - 12.2i)T + 1.36e3iT^{2} \)
41 \( 1 + 37.0T + 1.68e3T^{2} \)
43 \( 1 + (23.0 - 23.0i)T - 1.84e3iT^{2} \)
47 \( 1 + (4.22 + 4.22i)T + 2.20e3iT^{2} \)
53 \( 1 + (-23.0 + 23.0i)T - 2.80e3iT^{2} \)
59 \( 1 - 35.1iT - 3.48e3T^{2} \)
61 \( 1 + 73.3T + 3.72e3T^{2} \)
67 \( 1 + (43.6 + 43.6i)T + 4.48e3iT^{2} \)
71 \( 1 + 38.9T + 5.04e3T^{2} \)
73 \( 1 + (-67.6 + 67.6i)T - 5.32e3iT^{2} \)
79 \( 1 - 84.2iT - 6.24e3T^{2} \)
83 \( 1 + (-17.0 + 17.0i)T - 6.88e3iT^{2} \)
89 \( 1 - 96.4iT - 7.92e3T^{2} \)
97 \( 1 + (-98.7 - 98.7i)T + 9.40e3iT^{2} \)
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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

−10.24452331591698857125559733888, −9.252643690383617303950632742817, −8.274053550326252944235667670955, −7.57347283876820847962114066706, −6.46183567877580607282230334573, −5.82648897589426960219127356335, −5.06473076679881295662111620623, −3.78967834064815888281240469194, −3.38567978025979939526760254374, −1.45141535243788215207242054108, 0.41066773387750512394513124713, 1.77142605438774068832180527101, 2.87155137907801926303817441460, 4.01394383073164227702307682353, 4.98104532189124583016862115833, 5.90151250664601389873779666332, 6.65113372934056926256280314109, 7.47609623762054911378318912605, 8.724719307803181480388988813474, 9.392969639745904655562957798458

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