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Properties

Label 2-700-140.19-c1-0-33
Degree $2$
Conductor $700$
Sign $-0.389 + 0.920i$
Analytic cond. $5.58952$
Root an. cond. $2.36421$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36 + 0.366i)2-s + (−1.5 + 0.866i)3-s + (1.73 − i)4-s + (1.73 − 1.73i)6-s + (−2 + 1.73i)7-s + (−1.99 + 2i)8-s + (−0.866 + 0.5i)11-s + (−1.73 + 3i)12-s + 3.46·13-s + (2.09 − 3.09i)14-s + (1.99 − 3.46i)16-s + (−0.866 − 1.5i)17-s + (−2.59 + 4.5i)19-s + (1.50 − 4.33i)21-s + (0.999 − i)22-s + (−0.5 + 0.866i)23-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + (−0.866 + 0.499i)3-s + (0.866 − 0.5i)4-s + (0.707 − 0.707i)6-s + (−0.755 + 0.654i)7-s + (−0.707 + 0.707i)8-s + (−0.261 + 0.150i)11-s + (−0.499 + 0.866i)12-s + 0.960·13-s + (0.560 − 0.827i)14-s + (0.499 − 0.866i)16-s + (−0.210 − 0.363i)17-s + (−0.596 + 1.03i)19-s + (0.327 − 0.944i)21-s + (0.213 − 0.213i)22-s + (−0.104 + 0.180i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.389 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.389 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.389 + 0.920i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ -0.389 + 0.920i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 - 0.366i)T \)
5 \( 1 \)
7 \( 1 + (2 - 1.73i)T \)
good3 \( 1 + (1.5 - 0.866i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (0.866 - 0.5i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.46T + 13T^{2} \)
17 \( 1 + (0.866 + 1.5i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.59 - 4.5i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + (-0.866 - 1.5i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.59 + 1.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.46iT - 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (7.5 + 4.33i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.866 + 0.5i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.59 + 4.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.5 + 2.59i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.5 + 2.59i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 14iT - 71T^{2} \)
73 \( 1 + (4.33 + 7.5i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.79 - 4.5i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 13.8iT - 83T^{2} \)
89 \( 1 + (13.5 + 7.79i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 17.3T + 97T^{2} \)
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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

−10.25093408923117513068139029966, −9.360172500062220162399026246213, −8.594937280880762318089989446287, −7.67051299293385130172972427734, −6.42619331226204030803229455757, −5.93843318049884808087865252854, −5.04493490332622481562196194867, −3.46529966640841022350420741100, −2.00556098058206946839174090192, 0, 1.24570098572474117884266241376, 2.89891023692161438738605574849, 4.06016488768825871883223120311, 5.77036011178632091281173706890, 6.52493700935694254359418641056, 7.07831323330829588749992574098, 8.192795968226943265825803630642, 9.050946120714842197803539724584, 9.923969150727512271162371211515

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