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Properties

Label 2-700-140.19-c1-0-58
Degree $2$
Conductor $700$
Sign $-0.197 + 0.980i$
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  + (−1.29 − 0.569i)2-s + (2.62 − 1.51i)3-s + (1.35 + 1.47i)4-s + (−4.25 + 0.465i)6-s + (0.602 − 2.57i)7-s + (−0.908 − 2.67i)8-s + (3.08 − 5.33i)9-s + (−1.03 + 0.598i)11-s + (5.77 + 1.82i)12-s + 4.83·13-s + (−2.24 + 2.99i)14-s + (−0.349 + 3.98i)16-s + (−1.27 − 2.20i)17-s + (−7.02 + 5.15i)18-s + (−0.711 + 1.23i)19-s + ⋯
L(s)  = 1  + (−0.915 − 0.402i)2-s + (1.51 − 0.873i)3-s + (0.675 + 0.737i)4-s + (−1.73 + 0.190i)6-s + (0.227 − 0.973i)7-s + (−0.321 − 0.946i)8-s + (1.02 − 1.77i)9-s + (−0.312 + 0.180i)11-s + (1.66 + 0.525i)12-s + 1.34·13-s + (−0.600 + 0.799i)14-s + (−0.0873 + 0.996i)16-s + (−0.308 − 0.534i)17-s + (−1.65 + 1.21i)18-s + (−0.163 + 0.282i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.197 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.197 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.197 + 0.980i$
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: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ -0.197 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09160 - 1.33305i\)
\(L(\frac12)\) \(\approx\) \(1.09160 - 1.33305i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.29 + 0.569i)T \)
5 \( 1 \)
7 \( 1 + (-0.602 + 2.57i)T \)
good3 \( 1 + (-2.62 + 1.51i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (1.03 - 0.598i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.83T + 13T^{2} \)
17 \( 1 + (1.27 + 2.20i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.711 - 1.23i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.90 - 5.02i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.774T + 29T^{2} \)
31 \( 1 + (3.31 + 5.74i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.42 - 2.55i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.46iT - 41T^{2} \)
43 \( 1 - 1.38T + 43T^{2} \)
47 \( 1 + (-0.927 - 0.535i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.91 + 1.68i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.94 + 8.55i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-8.31 - 4.79i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.27 + 9.14i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 16.3iT - 71T^{2} \)
73 \( 1 + (-0.0535 - 0.0927i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-9.32 - 5.38i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 15.8iT - 83T^{2} \)
89 \( 1 + (-3.41 - 1.97i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 8.71T + 97T^{2} \)
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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

−9.913981519964679539884352428755, −9.289287750375081296191219944204, −8.330204266316852154126814051352, −7.83868696182748880602520218174, −7.18314145371857373665248707342, −6.24377966858446225694150934427, −4.05302166492428121999820922424, −3.30726012868797311130180022218, −2.10393976334260911251268095035, −1.11593106533242978677239715312, 1.91150074093338196653795327728, 2.84101045158562783466402563908, 4.07169843383701165145052336460, 5.36080783828307695351773078490, 6.37072970371210993056933646137, 7.67308692227595606632585252150, 8.557458532522356354369220004604, 8.717715024117032123558730006321, 9.492139704752678562898723332236, 10.56167196201511454147508101960

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