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Properties

Label 2-700-28.27-c1-0-43
Degree $2$
Conductor $700$
Sign $0.987 + 0.159i$
Analytic cond. $5.58952$
Root an. cond. $2.36421$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.20 + 0.736i)2-s + 2.79·3-s + (0.914 − 1.77i)4-s + (−3.37 + 2.06i)6-s + (−0.819 − 2.51i)7-s + (0.207 + 2.82i)8-s + 4.82·9-s + 1.47i·11-s + (2.55 − 4.97i)12-s − 5.83i·13-s + (2.84 + 2.43i)14-s + (−2.32 − 3.25i)16-s + 4.12i·17-s + (−5.82 + 3.55i)18-s + 5.11·19-s + ⋯
L(s)  = 1  + (−0.853 + 0.521i)2-s + 1.61·3-s + (0.457 − 0.889i)4-s + (−1.37 + 0.841i)6-s + (−0.309 − 0.950i)7-s + (0.0732 + 0.997i)8-s + 1.60·9-s + 0.444i·11-s + (0.738 − 1.43i)12-s − 1.61i·13-s + (0.759 + 0.650i)14-s + (−0.582 − 0.813i)16-s + 0.999i·17-s + (−1.37 + 0.838i)18-s + 1.17·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.159i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.987 + 0.159i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ 0.987 + 0.159i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.74003 - 0.139342i\)
\(L(\frac12)\) \(\approx\) \(1.74003 - 0.139342i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.20 - 0.736i)T \)
5 \( 1 \)
7 \( 1 + (0.819 + 2.51i)T \)
good3 \( 1 - 2.79T + 3T^{2} \)
11 \( 1 - 1.47iT - 11T^{2} \)
13 \( 1 + 5.83iT - 13T^{2} \)
17 \( 1 - 4.12iT - 17T^{2} \)
19 \( 1 - 5.11T + 19T^{2} \)
23 \( 1 + 2.08iT - 23T^{2} \)
29 \( 1 - 8.24T + 29T^{2} \)
31 \( 1 - 3.95T + 31T^{2} \)
37 \( 1 + 2.24T + 37T^{2} \)
41 \( 1 + 4.12iT - 41T^{2} \)
43 \( 1 + 2.94iT - 43T^{2} \)
47 \( 1 + 11.8T + 47T^{2} \)
53 \( 1 - 3.75T + 53T^{2} \)
59 \( 1 + 5.59T + 59T^{2} \)
61 \( 1 - 11.6iT - 61T^{2} \)
67 \( 1 + 12.7iT - 67T^{2} \)
71 \( 1 + 7.97iT - 71T^{2} \)
73 \( 1 - 12.3iT - 73T^{2} \)
79 \( 1 - 4.16iT - 79T^{2} \)
83 \( 1 - 2.79T + 83T^{2} \)
89 \( 1 - 12.3iT - 89T^{2} \)
97 \( 1 - 8.24iT - 97T^{2} \)
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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

−10.19612253891016615972563256658, −9.524183534185279169110575331977, −8.478118014545435605262004297765, −7.979080390685956177001864642581, −7.32651149340740192392257910319, −6.36686802319692966962407729936, −4.96563398625907770628048839578, −3.59360338560263395973965270952, −2.63538621375811174192601923985, −1.14479316599153236691346409504, 1.63040499840377052315207487754, 2.75280118581327215526016039516, 3.30026032892203428899712372624, 4.64349939924433268889413859066, 6.44335946864354633126448611837, 7.30420097799236808300388080572, 8.263446853184824134224900719808, 8.825747163584183903968696375041, 9.528052158187218248309340393677, 9.901624761336305409525613875514

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