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Properties

Label 2-700-7.2-c1-0-10
Degree $2$
Conductor $700$
Sign $-0.989 + 0.142i$
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  + (−0.866 − 1.5i)3-s + (0.209 + 2.63i)7-s + (−1.13 − 1.97i)11-s − 6.09·13-s + (−2.38 − 4.13i)17-s + (2.13 − 3.70i)19-s + (3.77 − 2.59i)21-s + (−0.447 + 0.774i)23-s − 5.19·27-s − 3.27·29-s + (2.13 + 3.70i)31-s + (−1.97 + 3.41i)33-s + (2.80 − 4.86i)37-s + (5.27 + 9.13i)39-s − 11.2·41-s + ⋯
L(s)  = 1  + (−0.499 − 0.866i)3-s + (0.0791 + 0.996i)7-s + (−0.342 − 0.594i)11-s − 1.68·13-s + (−0.579 − 1.00i)17-s + (0.490 − 0.849i)19-s + (0.823 − 0.566i)21-s + (−0.0932 + 0.161i)23-s − 1.00·27-s − 0.608·29-s + (0.383 + 0.664i)31-s + (−0.342 + 0.594i)33-s + (0.461 − 0.799i)37-s + (0.844 + 1.46i)39-s − 1.76·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.142i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 + 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.989 + 0.142i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ -0.989 + 0.142i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0338750 - 0.474397i\)
\(L(\frac12)\) \(\approx\) \(0.0338750 - 0.474397i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.209 - 2.63i)T \)
good3 \( 1 + (0.866 + 1.5i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (1.13 + 1.97i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 6.09T + 13T^{2} \)
17 \( 1 + (2.38 + 4.13i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.13 + 3.70i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.447 - 0.774i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.27T + 29T^{2} \)
31 \( 1 + (-2.13 - 3.70i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.80 + 4.86i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 11.2T + 41T^{2} \)
43 \( 1 + 6.50T + 43T^{2} \)
47 \( 1 + (1.07 - 1.86i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.70 + 6.41i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.13 + 3.70i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.774 - 1.34i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.95 - 12.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.5T + 71T^{2} \)
73 \( 1 + (1.07 + 1.86i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.137 + 0.238i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.67T + 83T^{2} \)
89 \( 1 + (-3.5 + 6.06i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 6.92T + 97T^{2} \)
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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

−9.879969789478787647082630039523, −9.248116554998555694967777629158, −8.212097007057137199555553781953, −7.21887912982848804789734777525, −6.64528495449458619963439348184, −5.46994001724909168804647473981, −4.88897885618587086111051803425, −3.06303144955831308829779934307, −2.04912056122037259148078230545, −0.24770195952458898628909719282, 1.96944228998277801746996768337, 3.62245920364958077236720398207, 4.59633971808217535230569560533, 5.12381119562401342509828192461, 6.43230099121010075670292879776, 7.44982948776909452537465269471, 8.094472272121280566945831966690, 9.624135349851606824640857597018, 10.03288612646902664732071216852, 10.62169996536594050979740864095

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