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Properties

Label 2-700-20.7-c1-0-4
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: $\chi_{700} (407, \cdot )$
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

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

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