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Properties

Label 2-700-140.67-c1-0-33
Degree $2$
Conductor $700$
Sign $0.902 - 0.430i$
Analytic cond. $5.58952$
Root an. cond. $2.36421$
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  + (−1.20 + 0.739i)2-s + (2.47 − 0.664i)3-s + (0.906 − 1.78i)4-s + (−2.49 + 2.63i)6-s + (−1.41 + 2.23i)7-s + (0.224 + 2.81i)8-s + (3.10 − 1.79i)9-s + (1.59 + 0.921i)11-s + (1.06 − 5.02i)12-s + (2.94 − 2.94i)13-s + (0.0455 − 3.74i)14-s + (−2.35 − 3.23i)16-s + (2.96 − 0.795i)17-s + (−2.41 + 4.45i)18-s + (2.66 + 4.61i)19-s + ⋯
L(s)  = 1  + (−0.852 + 0.522i)2-s + (1.43 − 0.383i)3-s + (0.453 − 0.891i)4-s + (−1.01 + 1.07i)6-s + (−0.533 + 0.846i)7-s + (0.0793 + 0.996i)8-s + (1.03 − 0.597i)9-s + (0.481 + 0.277i)11-s + (0.307 − 1.44i)12-s + (0.817 − 0.817i)13-s + (0.0121 − 0.999i)14-s + (−0.588 − 0.808i)16-s + (0.720 − 0.192i)17-s + (−0.570 + 1.05i)18-s + (0.611 + 1.05i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 - 0.430i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.902 - 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.902 - 0.430i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ 0.902 - 0.430i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64256 + 0.371250i\)
\(L(\frac12)\) \(\approx\) \(1.64256 + 0.371250i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.20 - 0.739i)T \)
5 \( 1 \)
7 \( 1 + (1.41 - 2.23i)T \)
good3 \( 1 + (-2.47 + 0.664i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (-1.59 - 0.921i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.94 + 2.94i)T - 13iT^{2} \)
17 \( 1 + (-2.96 + 0.795i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-2.66 - 4.61i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.654 + 2.44i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 6.35iT - 29T^{2} \)
31 \( 1 + (3.78 + 2.18i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.03 + 3.86i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 10.6T + 41T^{2} \)
43 \( 1 + (-1.02 - 1.02i)T + 43iT^{2} \)
47 \( 1 + (-0.785 - 0.210i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.699 + 2.61i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.11 - 3.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.00 + 10.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.856 + 3.19i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 10.5iT - 71T^{2} \)
73 \( 1 + (-0.529 - 1.97i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (3.95 + 6.85i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.227 - 0.227i)T + 83iT^{2} \)
89 \( 1 + (-3.75 + 2.16i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.196 + 0.196i)T + 97iT^{2} \)
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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

−10.08529478292944289462275905817, −9.300164621658838762934604605480, −8.825223951136063560479832765652, −7.969193708803843402669861407460, −7.39804777337610795976559266180, −6.25297768165720013552613111723, −5.45214784019173952052419484204, −3.58884791124101319253014426850, −2.63442808409138624049607028261, −1.40559202643516359127549458493, 1.24621386825213704116879224174, 2.71217059043217501217093232334, 3.58576604179124881810670123026, 4.21918360073036618876404166930, 6.28008057365453456086424578973, 7.32816732495371999304862793360, 7.940549272924150827863544535684, 9.065537111655067834647912677902, 9.270481705161573494219005107138, 10.13089499702205948862147395375

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