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Properties

Label 2-700-7.4-c1-0-4
Degree $2$
Conductor $700$
Sign $0.649 - 0.760i$
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  + (0.182 − 0.315i)3-s + (2.11 + 1.58i)7-s + (1.43 + 2.48i)9-s + (−0.682 + 1.18i)11-s − 2.63·13-s + (−1.11 + 1.93i)17-s + (3.11 + 5.39i)19-s + (0.886 − 0.378i)21-s + (−3.29 − 5.71i)23-s + 2.13·27-s + 5.50·29-s + (−2.25 + 3.89i)31-s + (0.248 + 0.430i)33-s + (1.04 + 1.81i)37-s + (−0.480 + 0.831i)39-s + ⋯
L(s)  = 1  + (0.105 − 0.182i)3-s + (0.799 + 0.600i)7-s + (0.477 + 0.827i)9-s + (−0.205 + 0.356i)11-s − 0.730·13-s + (−0.270 + 0.468i)17-s + (0.714 + 1.23i)19-s + (0.193 − 0.0825i)21-s + (−0.687 − 1.19i)23-s + 0.411·27-s + 1.02·29-s + (−0.404 + 0.700i)31-s + (0.0432 + 0.0749i)33-s + (0.172 + 0.298i)37-s + (−0.0769 + 0.133i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.649 - 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.649 - 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.649 - 0.760i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ 0.649 - 0.760i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48137 + 0.682463i\)
\(L(\frac12)\) \(\approx\) \(1.48137 + 0.682463i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 + (-2.11 - 1.58i)T \)
good3 \( 1 + (-0.182 + 0.315i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (0.682 - 1.18i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.63T + 13T^{2} \)
17 \( 1 + (1.11 - 1.93i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.11 - 5.39i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.29 + 5.71i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.50T + 29T^{2} \)
31 \( 1 + (2.25 - 3.89i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.04 - 1.81i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 9.32T + 41T^{2} \)
43 \( 1 + 1.86T + 43T^{2} \)
47 \( 1 + (-3.43 - 5.94i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.06 + 8.78i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.817 + 1.41i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.0197 + 0.0341i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.36 + 5.82i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.27T + 71T^{2} \)
73 \( 1 + (-2 + 3.46i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.66 + 4.61i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 14.7T + 83T^{2} \)
89 \( 1 + (0.433 + 0.750i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 16.0T + 97T^{2} \)
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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

−10.45299233588227656745565953816, −9.883347264821596996317081988236, −8.639546590930498857692726533694, −7.987178003698502894345325314783, −7.26439326013949639234928143284, −6.05284179402556510776862052393, −5.04708836169257973503410858296, −4.29074310486501131959158128032, −2.63014478093645052593305846409, −1.67591529041570361522640799497, 0.916569986102311121771317666912, 2.58220938862731626266526846038, 3.88895810229037990086264084520, 4.73774063958242966025832020863, 5.75835026353866284427160855531, 7.08842477259499207162996763515, 7.52443104972076199018803453152, 8.705128360428254927200706211196, 9.526555316571523868701206439944, 10.22906590647895695257922872730

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