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Properties

Label 2-700-20.3-c1-0-36
Degree $2$
Conductor $700$
Sign $-0.680 + 0.732i$
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  + (0.361 − 1.36i)2-s + (−1.26 + 1.26i)3-s + (−1.73 − 0.988i)4-s + (1.27 + 2.18i)6-s + (0.707 + 0.707i)7-s + (−1.97 + 2.02i)8-s − 0.204i·9-s − 2.11i·11-s + (3.45 − 0.950i)12-s + (−2.86 − 2.86i)13-s + (1.22 − 0.711i)14-s + (2.04 + 3.43i)16-s + (4.47 − 4.47i)17-s + (−0.280 − 0.0740i)18-s − 2.95·19-s + ⋯
L(s)  = 1  + (0.255 − 0.966i)2-s + (−0.730 + 0.730i)3-s + (−0.869 − 0.494i)4-s + (0.519 + 0.893i)6-s + (0.267 + 0.267i)7-s + (−0.699 + 0.714i)8-s − 0.0682i·9-s − 0.637i·11-s + (0.996 − 0.274i)12-s + (−0.794 − 0.794i)13-s + (0.326 − 0.190i)14-s + (0.511 + 0.859i)16-s + (1.08 − 1.08i)17-s + (−0.0660 − 0.0174i)18-s − 0.677·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.680 + 0.732i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.680 + 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.680 + 0.732i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ -0.680 + 0.732i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.332717 - 0.763138i\)
\(L(\frac12)\) \(\approx\) \(0.332717 - 0.763138i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.361 + 1.36i)T \)
5 \( 1 \)
7 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (1.26 - 1.26i)T - 3iT^{2} \)
11 \( 1 + 2.11iT - 11T^{2} \)
13 \( 1 + (2.86 + 2.86i)T + 13iT^{2} \)
17 \( 1 + (-4.47 + 4.47i)T - 17iT^{2} \)
19 \( 1 + 2.95T + 19T^{2} \)
23 \( 1 + (-3.94 + 3.94i)T - 23iT^{2} \)
29 \( 1 + 2.86iT - 29T^{2} \)
31 \( 1 + 6.01iT - 31T^{2} \)
37 \( 1 + (3.91 - 3.91i)T - 37iT^{2} \)
41 \( 1 + 7.98T + 41T^{2} \)
43 \( 1 + (-5.64 + 5.64i)T - 43iT^{2} \)
47 \( 1 + (4.30 + 4.30i)T + 47iT^{2} \)
53 \( 1 + (-0.561 - 0.561i)T + 53iT^{2} \)
59 \( 1 + 4.97T + 59T^{2} \)
61 \( 1 - 1.68T + 61T^{2} \)
67 \( 1 + (8.06 + 8.06i)T + 67iT^{2} \)
71 \( 1 - 0.610iT - 71T^{2} \)
73 \( 1 + (-3.23 - 3.23i)T + 73iT^{2} \)
79 \( 1 - 14.2T + 79T^{2} \)
83 \( 1 + (-5.51 + 5.51i)T - 83iT^{2} \)
89 \( 1 + 2.07iT - 89T^{2} \)
97 \( 1 + (-1.95 + 1.95i)T - 97iT^{2} \)
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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

−10.31110031544291262726343400701, −9.670899597866540227241935829978, −8.662111965535182752519191655183, −7.69901625945361384008496736373, −6.12518394952558492991469722916, −5.20291648851257963403209026726, −4.78127150502046754995450867784, −3.47780320384936432436205194856, −2.39160740750634140493926343391, −0.46463494162143809893765694369, 1.48495230487374791703257368352, 3.50440361593799523334081384353, 4.67928359501865441957568249650, 5.50740674114857734868975927629, 6.49615133705790005911172130762, 7.10678491689136485272724181658, 7.79710204649220387239401008470, 8.889350768128884848329821387801, 9.761562225601084989284685733545, 10.80893864885405293072818558821

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