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Properties

Label 2-700-140.67-c1-0-53
Degree $2$
Conductor $700$
Sign $0.281 + 0.959i$
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  + (−1.36 − 0.366i)2-s + (3.32 − 0.891i)3-s + (1.73 + i)4-s − 4.87·6-s + (−2.07 − 1.63i)7-s + (−1.99 − 2i)8-s + (7.68 − 4.43i)9-s + (6.65 + 1.78i)12-s + (2.23 + 3i)14-s + (1.99 + 3.46i)16-s + (−12.1 + 3.24i)18-s + (−8.37 − 3.60i)21-s + (1.94 − 7.25i)23-s + (−8.44 − 4.87i)24-s + (14.3 − 14.3i)27-s + (−1.95 − 4.91i)28-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (1.92 − 0.514i)3-s + (0.866 + 0.5i)4-s − 1.98·6-s + (−0.784 − 0.619i)7-s + (−0.707 − 0.707i)8-s + (2.56 − 1.47i)9-s + (1.92 + 0.514i)12-s + (0.597 + 0.801i)14-s + (0.499 + 0.866i)16-s + (−2.85 + 0.765i)18-s + (−1.82 − 0.786i)21-s + (0.405 − 1.51i)23-s + (−1.72 − 0.994i)24-s + (2.75 − 2.75i)27-s + (−0.369 − 0.929i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.281 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.281 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.281 + 0.959i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ 0.281 + 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44548 - 1.08232i\)
\(L(\frac12)\) \(\approx\) \(1.44548 - 1.08232i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.366i)T \)
5 \( 1 \)
7 \( 1 + (2.07 + 1.63i)T \)
good3 \( 1 + (-3.32 + 0.891i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 13iT^{2} \)
17 \( 1 + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.94 + 7.25i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 4.74iT - 29T^{2} \)
31 \( 1 + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 2.12T + 41T^{2} \)
43 \( 1 + (-6.43 - 6.43i)T + 43iT^{2} \)
47 \( 1 + (9.56 + 2.56i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.80 - 6.59i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.09 - 15.2i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.309 - 0.309i)T + 83iT^{2} \)
89 \( 1 + (-10.8 + 6.24i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 97iT^{2} \)
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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

−9.983088693795721214268960682537, −9.282977453577670346613116186548, −8.601387435511242322788667483301, −7.899288090207261254834505847738, −7.02064754014467423119919434568, −6.54525070645226428889736573584, −4.16486721836913346766063483187, −3.21586661150513275106539968849, −2.46947819733614277815502602629, −1.12389723073079538290891659309, 1.84517232430475735008966959230, 2.83344058812122120059814860481, 3.68332107955728956114163757644, 5.22072162131532459952386851737, 6.58125395828038220629414677658, 7.54214581014328701015586303652, 8.183310538416925903932875529703, 9.051632048266555032359814379239, 9.506944481552907704257471741095, 10.06465802602387777272158177534

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