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Properties

Label 2-700-28.27-c1-0-16
Degree $2$
Conductor $700$
Sign $0.573 - 0.818i$
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  + (−0.927 − 1.06i)2-s + 0.662·3-s + (−0.280 + 1.98i)4-s + (−0.613 − 0.707i)6-s + (2.35 + 1.19i)7-s + (2.37 − 1.53i)8-s − 2.56·9-s + 3.09i·11-s + (−0.185 + 1.31i)12-s + 4.66i·13-s + (−0.905 − 3.63i)14-s + (−3.84 − 1.11i)16-s + 2.04i·17-s + (2.37 + 2.73i)18-s − 5.60·19-s + ⋯
L(s)  = 1  + (−0.655 − 0.755i)2-s + 0.382·3-s + (−0.140 + 0.990i)4-s + (−0.250 − 0.288i)6-s + (0.891 + 0.453i)7-s + (0.839 − 0.543i)8-s − 0.853·9-s + 0.932i·11-s + (−0.0536 + 0.378i)12-s + 1.29i·13-s + (−0.242 − 0.970i)14-s + (−0.960 − 0.277i)16-s + 0.496i·17-s + (0.559 + 0.644i)18-s − 1.28·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.573 - 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.573 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.573 - 0.818i$
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: \((2,\ 700,\ (\ :1/2),\ 0.573 - 0.818i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.874797 + 0.455131i\)
\(L(\frac12)\) \(\approx\) \(0.874797 + 0.455131i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.927 + 1.06i)T \)
5 \( 1 \)
7 \( 1 + (-2.35 - 1.19i)T \)
good3 \( 1 - 0.662T + 3T^{2} \)
11 \( 1 - 3.09iT - 11T^{2} \)
13 \( 1 - 4.66iT - 13T^{2} \)
17 \( 1 - 2.04iT - 17T^{2} \)
19 \( 1 + 5.60T + 19T^{2} \)
23 \( 1 + 1.87iT - 23T^{2} \)
29 \( 1 - 3.56T + 29T^{2} \)
31 \( 1 + 8.74T + 31T^{2} \)
37 \( 1 - 3.70T + 37T^{2} \)
41 \( 1 - 8.48iT - 41T^{2} \)
43 \( 1 - 4.27iT - 43T^{2} \)
47 \( 1 - 0.290T + 47T^{2} \)
53 \( 1 - 9.49T + 53T^{2} \)
59 \( 1 - 8.05T + 59T^{2} \)
61 \( 1 - 6.45iT - 61T^{2} \)
67 \( 1 + 2.39iT - 67T^{2} \)
71 \( 1 + 9.65iT - 71T^{2} \)
73 \( 1 + 4.09iT - 73T^{2} \)
79 \( 1 - 1.35iT - 79T^{2} \)
83 \( 1 - 12.4T + 83T^{2} \)
89 \( 1 - 2.82iT - 89T^{2} \)
97 \( 1 + 6.14iT - 97T^{2} \)
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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

−10.67292605922517762200848832049, −9.596930593015942848135800825966, −8.842161454318004502096244648016, −8.329967114815739197661556110672, −7.40077150872757958254182265621, −6.29495246181828791292242684521, −4.79171844672115497830936933850, −3.97983731436953923111103598328, −2.48853792048795763951643506625, −1.79110220731301688216028448012, 0.60322614257590215816289066213, 2.31244676835779420694027967051, 3.80280256103064737391284400516, 5.24053440007576711416778547458, 5.77690568842725059492717851304, 7.01752693295375863200607709148, 7.935840055609105681708016513813, 8.458556644658684274552484850951, 9.109721099599524075939878484016, 10.39247748894433894658540866195

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