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Properties

Label 2-700-140.27-c0-0-1
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

\(L(\frac{1}{2})\) \(\approx\) \(0.5867867760\)
\(L(\frac12)\) \(\approx\) \(0.5867867760\)
\(L(1)\) 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

−10.73098792285850473462189574181, −9.873750436357749173367187532386, −9.062762101882822741284211128927, −8.431442871835027251479942827423, −7.35232489365937480203590558845, −6.72670947749367994039517362325, −5.50586403046430644841154787085, −5.02992254519506161452014467640, −3.22320742186180687134708825688, −1.79710515612962831056229060091, 0.846763681165843152092747064034, 2.66049536488616773387402090552, 3.62684846329334024325044322582, 4.55594170247509726027221479847, 6.32901388526955899998722511304, 6.92689804270752575492875530616, 7.986476815199382154730814820442, 8.896642961291530397978538190496, 9.692394288438552140145991874321, 10.23809181413618453542343597327

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