Properties

Label 2-700-140.27-c0-0-5
Degree $2$
Conductor $700$
Sign $0.229 + 0.973i$
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  + (0.707 − 0.707i)2-s − 1.00i·4-s + (0.707 − 0.707i)7-s + (−0.707 − 0.707i)8-s + i·9-s − 1.00i·14-s − 1.00·16-s + (0.707 + 0.707i)18-s + (−1.41 − 1.41i)23-s + (−0.707 − 0.707i)28-s + 2i·29-s + (−0.707 + 0.707i)32-s + 1.00·36-s + (1.41 + 1.41i)43-s − 2.00·46-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00i·4-s + (0.707 − 0.707i)7-s + (−0.707 − 0.707i)8-s + i·9-s − 1.00i·14-s − 1.00·16-s + (0.707 + 0.707i)18-s + (−1.41 − 1.41i)23-s + (−0.707 − 0.707i)28-s + 2i·29-s + (−0.707 + 0.707i)32-s + 1.00·36-s + (1.41 + 1.41i)43-s − 2.00·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.229 + 0.973i$
Analytic conductor: \(0.349345\)
Root analytic conductor: \(0.591054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :0),\ 0.229 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.369092322\)
\(L(\frac12)\) \(\approx\) \(1.369092322\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 \)
7 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 - iT^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
29 \( 1 - 2iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + T^{2} \)
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

−10.65776646749660624276229901750, −10.04061072221709942574238249893, −8.810051274616026849948882966244, −7.86621014943965012687747934930, −6.87180082471471411966888682280, −5.73364110536209345822163853664, −4.76388419558164005128833273915, −4.11917056743580946188367879326, −2.72053426732590762748032989463, −1.54486986127973143592217368990, 2.21516036612903868742917252508, 3.58245483247344544841672958947, 4.45522278145457614265064462552, 5.70007574692904338507772889874, 6.10386607404840605350932034513, 7.36591469182560150197083690031, 8.065485621490381337846663777782, 8.973922913065695820493529002068, 9.745418629209944116986820700641, 11.16783967071500255630090898589

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