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Properties

Label 2-700-28.19-c1-0-1
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

\(L(1)\) \(\approx\) \(0.0782375 + 0.0230730i\)
\(L(\frac12)\) \(\approx\) \(0.0782375 + 0.0230730i\)
\(L(\frac{3}{2})\) 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

−10.78533865934138978758871078987, −10.02030265666431763261175748987, −8.740834115982041560187400129272, −7.84353711955032593741451426772, −7.18143657517573119109013869672, −6.17887440451359413579161697720, −4.93703815883524410958874646378, −4.00438915953378249908170777007, −2.34884787402766119329566969672, −1.39665161026020432350178332210, 0.05216694723263177385890307820, 2.83153165793376765578690330858, 4.50187091487278627424846641118, 4.92545968432631411043060869969, 5.78717150538435654247947067448, 6.57354695378038961254524633951, 7.86578791857027914882602087724, 8.864412163142602567491151821995, 9.285218709643996299509484001421, 10.35062235677126706803270671185

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