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Properties

Label 2-700-35.27-c1-0-8
Degree $2$
Conductor $700$
Sign $0.839 + 0.543i$
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  + (0.386 + 0.386i)3-s + (−2.64 − 0.0564i)7-s − 2.70i·9-s + 1.70·11-s + (0.386 + 0.386i)13-s + (4.79 − 4.79i)17-s + 5.95·19-s + (−0.999 − 1.04i)21-s + (2.70 − 2.70i)23-s + (2.20 − 2.20i)27-s − 5.70i·29-s + 8.03i·31-s + (0.657 + 0.657i)33-s + (−2.70 − 2.70i)37-s + 0.298i·39-s + ⋯
L(s)  = 1  + (0.223 + 0.223i)3-s + (−0.999 − 0.0213i)7-s − 0.900i·9-s + 0.513·11-s + (0.107 + 0.107i)13-s + (1.16 − 1.16i)17-s + 1.36·19-s + (−0.218 − 0.227i)21-s + (0.563 − 0.563i)23-s + (0.423 − 0.423i)27-s − 1.05i·29-s + 1.44i·31-s + (0.114 + 0.114i)33-s + (−0.444 − 0.444i)37-s + 0.0477i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.839 + 0.543i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.839 + 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.839 + 0.543i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (657, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ 0.839 + 0.543i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46243 - 0.432347i\)
\(L(\frac12)\) \(\approx\) \(1.46243 - 0.432347i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 + (2.64 + 0.0564i)T \)
good3 \( 1 + (-0.386 - 0.386i)T + 3iT^{2} \)
11 \( 1 - 1.70T + 11T^{2} \)
13 \( 1 + (-0.386 - 0.386i)T + 13iT^{2} \)
17 \( 1 + (-4.79 + 4.79i)T - 17iT^{2} \)
19 \( 1 - 5.95T + 19T^{2} \)
23 \( 1 + (-2.70 + 2.70i)T - 23iT^{2} \)
29 \( 1 + 5.70iT - 29T^{2} \)
31 \( 1 - 8.03iT - 31T^{2} \)
37 \( 1 + (2.70 + 2.70i)T + 37iT^{2} \)
41 \( 1 - 5.95iT - 41T^{2} \)
43 \( 1 + (-5 + 5i)T - 43iT^{2} \)
47 \( 1 + (3.24 - 3.24i)T - 47iT^{2} \)
53 \( 1 + (5 - 5i)T - 53iT^{2} \)
59 \( 1 + 5.95T + 59T^{2} \)
61 \( 1 + 11.9iT - 61T^{2} \)
67 \( 1 + (5 + 5i)T + 67iT^{2} \)
71 \( 1 - 7.40T + 71T^{2} \)
73 \( 1 + (1.81 + 1.81i)T + 73iT^{2} \)
79 \( 1 + 0.298iT - 79T^{2} \)
83 \( 1 + (-4.13 - 4.13i)T + 83iT^{2} \)
89 \( 1 + 2.08T + 89T^{2} \)
97 \( 1 + (1.15 - 1.15i)T - 97iT^{2} \)
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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

−10.07530973989221724018646087057, −9.491515274171958124574170106333, −8.974728373169916163488781412315, −7.67034321109109908785657360803, −6.81721288628008015278635200783, −6.01176113351831736819504528409, −4.86848929035937699251008857289, −3.53823372381464758069731016411, −2.98330015600952953633873626733, −0.907129716014309692566401199852, 1.41042005595908317200264272396, 2.94641993904798019925147598790, 3.79445291350391243776887792972, 5.24794781219584212175811859232, 6.03641005847215124443103080198, 7.16090866177728280349942307432, 7.81777947810573298928503256634, 8.848355612890790745247603763114, 9.705857684038243121485226449982, 10.39642546907562965922048942831

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