Properties

Label 2-700-140.47-c0-0-1
Degree $2$
Conductor $700$
Sign $-0.104 - 0.994i$
Analytic cond. $0.349345$
Root an. cond. $0.591054$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.258 + 0.965i)2-s + (1.67 + 0.448i)3-s + (−0.866 + 0.499i)4-s + 1.73i·6-s + (−0.965 − 0.258i)7-s + (−0.707 − 0.707i)8-s + (1.73 + 1.00i)9-s + (−1.67 + 0.448i)12-s i·14-s + (0.500 − 0.866i)16-s + (−0.517 + 1.93i)18-s + (−1.50 − 0.866i)21-s + (−0.965 + 0.258i)23-s + (−0.866 − 1.5i)24-s + (1.22 + 1.22i)27-s + (0.965 − 0.258i)28-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (1.67 + 0.448i)3-s + (−0.866 + 0.499i)4-s + 1.73i·6-s + (−0.965 − 0.258i)7-s + (−0.707 − 0.707i)8-s + (1.73 + 1.00i)9-s + (−1.67 + 0.448i)12-s i·14-s + (0.500 − 0.866i)16-s + (−0.517 + 1.93i)18-s + (−1.50 − 0.866i)21-s + (−0.965 + 0.258i)23-s + (−0.866 − 1.5i)24-s + (1.22 + 1.22i)27-s + (0.965 − 0.258i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.104 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.104 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.104 - 0.994i$
Analytic conductor: \(0.349345\)
Root analytic conductor: \(0.591054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :0),\ -0.104 - 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.480563750\)
\(L(\frac12)\) \(\approx\) \(1.480563750\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
5 \( 1 \)
7 \( 1 + (0.965 + 0.258i)T \)
good3 \( 1 + (-1.67 - 0.448i)T + (0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (0.866 + 0.5i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
29 \( 1 + iT - T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.866 - 0.5i)T^{2} \)
41 \( 1 + 1.73iT - T^{2} \)
43 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
47 \( 1 + (0.866 - 0.5i)T^{2} \)
53 \( 1 + (0.866 + 0.5i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.866 - 0.5i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
89 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 - iT^{2} \)
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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

−10.32262794269900972855049603541, −9.730425112141619918342764808796, −9.028166050649884059902474546420, −8.288348572640646917278545254357, −7.51383587706984835631540816309, −6.70306538628707702238085900301, −5.53113956986450807508410709260, −4.13066248015652618089986258404, −3.65369681191672722449734068898, −2.51864350358236475832139659836, 1.69905516496417531788434697241, 2.81786966784735972763852593500, 3.40492931718971685659854767103, 4.46665778558728879458703850106, 5.97646501476854815568773934564, 7.03962558035291243053229804233, 8.224798038958640056660965339193, 8.768542935287894774159792402570, 9.701374829161424181296734590891, 10.04137596140903311799905789120

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