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Properties

Label 2-700-35.19-c4-0-16
Degree $2$
Conductor $700$
Sign $-0.475 - 0.879i$
Analytic cond. $72.3589$
Root an. cond. $8.50640$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.79 + 6.58i)3-s + (30.9 + 38.0i)7-s + (11.6 − 20.1i)9-s + (100. + 174. i)11-s − 86.3·13-s + (14.2 + 24.6i)17-s + (32.9 + 19.0i)19-s + (−132. + 347. i)21-s + (511. + 295. i)23-s + 792.·27-s − 997.·29-s + (−24.7 + 14.3i)31-s + (−767. + 1.32e3i)33-s + (1.02e3 + 590. i)37-s + (−328. − 568. i)39-s + ⋯
L(s)  = 1  + (0.422 + 0.731i)3-s + (0.630 + 0.775i)7-s + (0.143 − 0.248i)9-s + (0.834 + 1.44i)11-s − 0.511·13-s + (0.0492 + 0.0852i)17-s + (0.0912 + 0.0526i)19-s + (−0.301 + 0.788i)21-s + (0.966 + 0.558i)23-s + 1.08·27-s − 1.18·29-s + (−0.0258 + 0.0148i)31-s + (−0.704 + 1.22i)33-s + (0.746 + 0.431i)37-s + (−0.215 − 0.373i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.475 - 0.879i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.475 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.475 - 0.879i$
Analytic conductor: \(72.3589\)
Root analytic conductor: \(8.50640\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :2),\ -0.475 - 0.879i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.757640268\)
\(L(\frac12)\) \(\approx\) \(2.757640268\)
\(L(3)\) 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 \( 1 + (-30.9 - 38.0i)T \)
good3 \( 1 + (-3.79 - 6.58i)T + (-40.5 + 70.1i)T^{2} \)
11 \( 1 + (-100. - 174. i)T + (-7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + 86.3T + 2.85e4T^{2} \)
17 \( 1 + (-14.2 - 24.6i)T + (-4.17e4 + 7.23e4i)T^{2} \)
19 \( 1 + (-32.9 - 19.0i)T + (6.51e4 + 1.12e5i)T^{2} \)
23 \( 1 + (-511. - 295. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + 997.T + 7.07e5T^{2} \)
31 \( 1 + (24.7 - 14.3i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + (-1.02e3 - 590. i)T + (9.37e5 + 1.62e6i)T^{2} \)
41 \( 1 - 571. iT - 2.82e6T^{2} \)
43 \( 1 + 1.29e3iT - 3.41e6T^{2} \)
47 \( 1 + (-975. + 1.68e3i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (362. - 209. i)T + (3.94e6 - 6.83e6i)T^{2} \)
59 \( 1 + (-1.66e3 + 961. i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-1.88e3 - 1.09e3i)T + (6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (5.44e3 - 3.14e3i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 1.59e3T + 2.54e7T^{2} \)
73 \( 1 + (1.75e3 + 3.03e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (5.19e3 - 8.99e3i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 1.12e4T + 4.74e7T^{2} \)
89 \( 1 + (-6.30e3 - 3.64e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + 2.89e3T + 8.85e7T^{2} \)
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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

−9.821119508800855304789994116615, −9.406363868714169905794569922514, −8.678011736682664966137740504961, −7.53867071239848208380773376557, −6.74614030247883696958108558720, −5.43382686096420510889026910744, −4.60177571891533713185657889877, −3.76287753246848777318467220216, −2.48806375483300385363383751614, −1.39369672849662514925908610367, 0.64271545147724007825936572053, 1.52469459011882792756359923605, 2.78501900139464078988482431107, 3.93323609478957426599219222041, 4.98910890027246688328813660305, 6.17124010028884561570905092556, 7.14321678168767358320378392996, 7.76330222511176730304613977723, 8.591942523332277927092119567508, 9.407426326371909797625376151659

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