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Properties

Label 2-700-28.19-c1-0-17
Degree $2$
Conductor $700$
Sign $-0.839 + 0.542i$
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  + (0.285 + 1.38i)2-s + (1.29 + 2.24i)3-s + (−1.83 + 0.791i)4-s + (−2.74 + 2.44i)6-s + (−0.603 + 2.57i)7-s + (−1.62 − 2.31i)8-s + (−1.87 + 3.23i)9-s + (−3.12 + 1.80i)11-s + (−4.16 − 3.10i)12-s − 0.818i·13-s + (−3.74 − 0.100i)14-s + (2.74 − 2.90i)16-s + (6.40 − 3.69i)17-s + (−5.02 − 1.66i)18-s + (−1.65 + 2.86i)19-s + ⋯
L(s)  = 1  + (0.202 + 0.979i)2-s + (0.749 + 1.29i)3-s + (−0.918 + 0.395i)4-s + (−1.11 + 0.996i)6-s + (−0.228 + 0.973i)7-s + (−0.573 − 0.819i)8-s + (−0.623 + 1.07i)9-s + (−0.940 + 0.543i)11-s + (−1.20 − 0.895i)12-s − 0.226i·13-s + (−0.999 − 0.0267i)14-s + (0.686 − 0.727i)16-s + (1.55 − 0.896i)17-s + (−1.18 − 0.392i)18-s + (−0.379 + 0.656i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.839 + 0.542i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.839 + 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.839 + 0.542i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (551, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ -0.839 + 0.542i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.447574 - 1.51766i\)
\(L(\frac12)\) \(\approx\) \(0.447574 - 1.51766i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.285 - 1.38i)T \)
5 \( 1 \)
7 \( 1 + (0.603 - 2.57i)T \)
good3 \( 1 + (-1.29 - 2.24i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (3.12 - 1.80i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.818iT - 13T^{2} \)
17 \( 1 + (-6.40 + 3.69i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.65 - 2.86i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.19 + 1.26i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.04T + 29T^{2} \)
31 \( 1 + (-0.955 - 1.65i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.58 - 6.20i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.65iT - 41T^{2} \)
43 \( 1 - 2.39iT - 43T^{2} \)
47 \( 1 + (0.667 - 1.15i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.905 - 1.56i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.955 - 1.65i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (8.46 + 4.88i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.02 + 4.63i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.38iT - 71T^{2} \)
73 \( 1 + (-6.40 + 3.69i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.70 - 3.87i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.4T + 83T^{2} \)
89 \( 1 + (-9.19 - 5.30i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 7.32iT - 97T^{2} \)
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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

−10.45652346717413777776667761842, −9.819698394861547986572554577265, −9.276202545364991508564284991741, −8.250932357777219965125217432169, −7.82816457172031431082950294875, −6.41765746557327719467149736888, −5.30613552033220133970408967328, −4.83025645087452880323168198987, −3.55838744884658534707972756805, −2.76723622436696787808720647064, 0.72412543604467231218731059351, 1.93810395077069187582148397279, 3.04891883740362443528670831744, 3.91061747377419884018025230080, 5.35792370923609227795409161867, 6.44250190991194990184983161642, 7.60574666926122211948251277837, 8.120599810753071213178944101657, 9.049570826545697830817702886836, 10.18410131991922792114443684460

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