Properties

Label 2-700-35.9-c1-0-8
Degree $2$
Conductor $700$
Sign $0.830 + 0.556i$
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  + (2.59 − 1.5i)3-s + (2.59 + 0.5i)7-s + (3 − 5.19i)9-s + (1 + 1.73i)11-s + 6i·13-s + (1.73 − i)17-s + (7.5 − 2.59i)21-s + (−7.79 − 4.5i)23-s − 9i·27-s − 3·29-s + (−1 − 1.73i)31-s + (5.19 + 3i)33-s + (−6.92 − 4i)37-s + (9 + 15.5i)39-s + 5·41-s + ⋯
L(s)  = 1  + (1.49 − 0.866i)3-s + (0.981 + 0.188i)7-s + (1 − 1.73i)9-s + (0.301 + 0.522i)11-s + 1.66i·13-s + (0.420 − 0.242i)17-s + (1.63 − 0.566i)21-s + (−1.62 − 0.938i)23-s − 1.73i·27-s − 0.557·29-s + (−0.179 − 0.311i)31-s + (0.904 + 0.522i)33-s + (−1.13 − 0.657i)37-s + (1.44 + 2.49i)39-s + 0.780·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 + 0.556i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.830 + 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.830 + 0.556i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ 0.830 + 0.556i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.58992 - 0.787553i\)
\(L(\frac12)\) \(\approx\) \(2.58992 - 0.787553i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.59 - 0.5i)T \)
good3 \( 1 + (-2.59 + 1.5i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 6iT - 13T^{2} \)
17 \( 1 + (-1.73 + i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (7.79 + 4.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.92 + 4i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 5T + 41T^{2} \)
43 \( 1 + iT - 43T^{2} \)
47 \( 1 + (6.92 + 4i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.46 - 2i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4 + 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.59 - 1.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (12.1 - 7i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - iT - 83T^{2} \)
89 \( 1 + (-6.5 + 11.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10iT - 97T^{2} \)
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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

−10.11512271689173922218485387170, −9.175052635047281800735947634756, −8.659146693144733000392518224338, −7.76098436563763344394501535172, −7.17221391813796075269955755591, −6.18295792692781301525212610270, −4.59818711338232673591791562115, −3.72783207606035734724301947734, −2.21874116559189075748288815535, −1.70756452577681252597114183710, 1.72196293071766099245710907694, 3.10169379256012402887933854753, 3.78097153947895385409843541218, 4.87661561592478920467818965468, 5.84711342882042746286338130168, 7.63793255811454605600635851568, 7.992600297852710622843356644228, 8.706178860224578912191709691762, 9.675723734676498759228744250302, 10.34880614579111213119087344008

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