Properties

Label 2-700-7.4-c1-0-5
Degree $2$
Conductor $700$
Sign $0.486 - 0.873i$
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 + (2.38 + 1.13i)7-s + (2.63 − 4.56i)11-s + 2.62·13-s + (−0.209 + 0.362i)17-s + (−1.63 − 2.83i)19-s + (−3.77 + 2.59i)21-s + (3.91 + 6.77i)23-s − 5.19·27-s + 4.27·29-s + (−1.63 + 2.83i)31-s + (4.56 + 7.91i)33-s + (4.98 + 8.63i)37-s + (−2.27 + 3.94i)39-s − 3.72·41-s + ⋯
L(s)  = 1  + (−0.499 + 0.866i)3-s + (0.902 + 0.429i)7-s + (0.795 − 1.37i)11-s + 0.728·13-s + (−0.0507 + 0.0879i)17-s + (−0.375 − 0.650i)19-s + (−0.823 + 0.566i)21-s + (0.815 + 1.41i)23-s − 1.00·27-s + 0.793·29-s + (−0.294 + 0.509i)31-s + (0.795 + 1.37i)33-s + (0.819 + 1.41i)37-s + (−0.364 + 0.630i)39-s − 0.581·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.486 - 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.486 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.486 - 0.873i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ 0.486 - 0.873i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31050 + 0.770506i\)
\(L(\frac12)\) \(\approx\) \(1.31050 + 0.770506i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.38 - 1.13i)T \)
good3 \( 1 + (0.866 - 1.5i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-2.63 + 4.56i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.62T + 13T^{2} \)
17 \( 1 + (0.209 - 0.362i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.63 + 2.83i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.91 - 6.77i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.27T + 29T^{2} \)
31 \( 1 + (1.63 - 2.83i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.98 - 8.63i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.72T + 41T^{2} \)
43 \( 1 + 2.15T + 43T^{2} \)
47 \( 1 + (3.25 + 5.63i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.83 + 4.91i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.63 + 2.83i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.77 - 11.7i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.76 - 3.04i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.54T + 71T^{2} \)
73 \( 1 + (3.25 - 5.63i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.63 + 6.30i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.40T + 83T^{2} \)
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

−10.75248110888651546977575615766, −9.821969975305646413602158627860, −8.761994205973019387214281993027, −8.352322369462669149786541810102, −6.97846634316705582967579005715, −5.88378195623161211641256070460, −5.17911708521264930780621539019, −4.23256867405151727780494334325, −3.16752585559258338742474332763, −1.37418934929991098085460143080, 1.06355585062648644404595131153, 2.07469347460716453669495629770, 3.96224881433107441035677121778, 4.75092454294923231241687925444, 6.06986121767843975647920051884, 6.81086553420221683922241001319, 7.50939748208063404432810372787, 8.444159187043064353707137507773, 9.439711202994429064574662006937, 10.46655079931175382395858448431

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