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Properties

Label 2-700-140.83-c0-0-3
Degree $2$
Conductor $700$
Sign $0.923 + 0.382i$
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 + (−0.866 − 0.499i)4-s + (−0.707 − 0.707i)7-s + (0.707 − 0.707i)8-s − i·9-s − 1.73i·11-s + (0.866 − 0.500i)14-s + (0.500 + 0.866i)16-s + (0.965 + 0.258i)18-s + (1.67 + 0.448i)22-s + (−0.707 + 0.707i)23-s + (0.258 + 0.965i)28-s + i·29-s + (−0.965 + 0.258i)32-s + (−0.499 + 0.866i)36-s + (1.22 − 1.22i)37-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s + (−0.866 − 0.499i)4-s + (−0.707 − 0.707i)7-s + (0.707 − 0.707i)8-s − i·9-s − 1.73i·11-s + (0.866 − 0.500i)14-s + (0.500 + 0.866i)16-s + (0.965 + 0.258i)18-s + (1.67 + 0.448i)22-s + (−0.707 + 0.707i)23-s + (0.258 + 0.965i)28-s + i·29-s + (−0.965 + 0.258i)32-s + (−0.499 + 0.866i)36-s + (1.22 − 1.22i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.923 + 0.382i$
Analytic conductor: \(0.349345\)
Root analytic conductor: \(0.591054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (643, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :0),\ 0.923 + 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6488993703\)
\(L(\frac12)\) \(\approx\) \(0.6488993703\)
\(L(1)\) 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 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + iT^{2} \)
11 \( 1 + 1.73iT - T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
29 \( 1 - iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
71 \( 1 - 1.73iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - 1.73T + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + T^{2} \)
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

−10.44028159250756203437251470241, −9.469915163385092177126844770450, −8.920611488146719038590562282502, −7.942285155899984874858205591521, −7.03756983895466551412712725269, −6.18621474772804581354988699248, −5.60968098006384188437277417153, −4.06457686784052564798066296956, −3.34783353415750436156921309837, −0.804929165529084770355553265484, 1.98805394384945882006328897110, 2.73231768835505107283855656580, 4.23334257377945410391829502847, 4.95247077873262036031376117498, 6.23578288166587369387614805734, 7.51444128250068842465806506959, 8.214114294649035784754420745721, 9.328888631941133333912872975829, 9.889744785430964022676723467245, 10.54801336012114048920984632332

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