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Properties

Label 2-700-20.7-c1-0-53
Degree $2$
Conductor $700$
Sign $0.628 - 0.777i$
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  + (0.622 − 1.26i)2-s + (−2.09 − 2.09i)3-s + (−1.22 − 1.58i)4-s + (−3.96 + 1.35i)6-s + (0.707 − 0.707i)7-s + (−2.76 + 0.572i)8-s + 5.78i·9-s + 0.214i·11-s + (−0.744 + 5.88i)12-s + (−4.29 + 4.29i)13-s + (−0.457 − 1.33i)14-s + (−0.996 + 3.87i)16-s + (−2.00 − 2.00i)17-s + (7.34 + 3.60i)18-s − 0.877·19-s + ⋯
L(s)  = 1  + (0.440 − 0.897i)2-s + (−1.21 − 1.21i)3-s + (−0.612 − 0.790i)4-s + (−1.61 + 0.554i)6-s + (0.267 − 0.267i)7-s + (−0.979 + 0.202i)8-s + 1.92i·9-s + 0.0648i·11-s + (−0.214 + 1.69i)12-s + (−1.19 + 1.19i)13-s + (−0.122 − 0.357i)14-s + (−0.249 + 0.968i)16-s + (−0.487 − 0.487i)17-s + (1.73 + 0.848i)18-s − 0.201·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.628 - 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.628 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.628 - 0.777i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ 0.628 - 0.777i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0522855 + 0.0249763i\)
\(L(\frac12)\) \(\approx\) \(0.0522855 + 0.0249763i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.622 + 1.26i)T \)
5 \( 1 \)
7 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (2.09 + 2.09i)T + 3iT^{2} \)
11 \( 1 - 0.214iT - 11T^{2} \)
13 \( 1 + (4.29 - 4.29i)T - 13iT^{2} \)
17 \( 1 + (2.00 + 2.00i)T + 17iT^{2} \)
19 \( 1 + 0.877T + 19T^{2} \)
23 \( 1 + (0.0902 + 0.0902i)T + 23iT^{2} \)
29 \( 1 - 4.03iT - 29T^{2} \)
31 \( 1 + 8.60iT - 31T^{2} \)
37 \( 1 + (-1.29 - 1.29i)T + 37iT^{2} \)
41 \( 1 - 2.91T + 41T^{2} \)
43 \( 1 + (2.06 + 2.06i)T + 43iT^{2} \)
47 \( 1 + (4.88 - 4.88i)T - 47iT^{2} \)
53 \( 1 + (-2.77 + 2.77i)T - 53iT^{2} \)
59 \( 1 + 10.4T + 59T^{2} \)
61 \( 1 + 8.32T + 61T^{2} \)
67 \( 1 + (0.555 - 0.555i)T - 67iT^{2} \)
71 \( 1 - 1.75iT - 71T^{2} \)
73 \( 1 + (4.18 - 4.18i)T - 73iT^{2} \)
79 \( 1 + 12.4T + 79T^{2} \)
83 \( 1 + (-5.30 - 5.30i)T + 83iT^{2} \)
89 \( 1 + 12.7iT - 89T^{2} \)
97 \( 1 + (-2.58 - 2.58i)T + 97iT^{2} \)
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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

−9.983993078650899872967408339761, −9.074435028288306827647202825666, −7.70500221814344123302924906933, −6.87294207287652687709491120563, −6.07872537519347773693776131417, −5.03861181240412676251782621718, −4.34343222207263778616745556200, −2.48652827553978995639038183885, −1.51621547852878081158542555743, −0.03059486263527961543714318555, 3.06743035188369678627209021341, 4.30056859145328635128329157272, 4.97672938386598329929266291335, 5.66217978324284152369938676252, 6.45560487140030395277878676035, 7.57668311356499326314110202280, 8.592518695485613254540510396061, 9.552816804458739201398033511153, 10.31655985240910606357563569024, 11.11037396360229145172233389198

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