Properties

Label 2-700-140.83-c0-0-4
Degree $2$
Conductor $700$
Sign $-0.189 + 0.981i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.189 + 0.981i$
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: \((2,\ 700,\ (\ :0),\ -0.189 + 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.021141299\)
\(L(\frac12)\) \(\approx\) \(1.021141299\)
\(L(1)\) 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

−10.68027621927717739353328126882, −9.591093193114020833478290211755, −8.641963041636095553842917747629, −8.424465113657670869833415313398, −6.63841932486733731973910769972, −5.71383856550288855664704992372, −4.90036714821025696929582212521, −3.59351526258679245455076876853, −2.81768883972257546820327323100, −1.21666096609023860472941152110, 1.99890243323828732158319712101, 3.81019901284444562517276017224, 4.76782229060643164186192278365, 5.27610741289750045750402563679, 6.71976293802846803509204039572, 7.47684906538455315705803426497, 7.906552358505447036317718354883, 9.054392002372552985228711573342, 9.964647190554390981696782640025, 10.75211599348353863405258966867

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