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Properties

Label 2-700-140.59-c1-0-10
Degree $2$
Conductor $700$
Sign $0.999 - 0.0422i$
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  + (−0.626 + 1.26i)2-s + (−2.59 − 1.49i)3-s + (−1.21 − 1.58i)4-s + (3.52 − 2.35i)6-s + (−1.65 + 2.06i)7-s + (2.77 − 0.546i)8-s + (2.99 + 5.18i)9-s + (−1.93 − 1.11i)11-s + (0.774 + 5.94i)12-s − 3.17·13-s + (−1.57 − 3.39i)14-s + (−1.04 + 3.86i)16-s + (−1.72 + 2.98i)17-s + (−8.45 + 0.548i)18-s + (−1.02 − 1.77i)19-s + ⋯
L(s)  = 1  + (−0.442 + 0.896i)2-s + (−1.49 − 0.865i)3-s + (−0.607 − 0.794i)4-s + (1.43 − 0.960i)6-s + (−0.627 + 0.778i)7-s + (0.981 − 0.193i)8-s + (0.998 + 1.72i)9-s + (−0.584 − 0.337i)11-s + (0.223 + 1.71i)12-s − 0.879·13-s + (−0.420 − 0.907i)14-s + (−0.261 + 0.965i)16-s + (−0.417 + 0.723i)17-s + (−1.99 + 0.129i)18-s + (−0.235 − 0.407i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0422i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0422i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.999 - 0.0422i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ 0.999 - 0.0422i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.415286 + 0.00877363i\)
\(L(\frac12)\) \(\approx\) \(0.415286 + 0.00877363i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.626 - 1.26i)T \)
5 \( 1 \)
7 \( 1 + (1.65 - 2.06i)T \)
good3 \( 1 + (2.59 + 1.49i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (1.93 + 1.11i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.17T + 13T^{2} \)
17 \( 1 + (1.72 - 2.98i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.02 + 1.77i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.33 + 2.30i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.38T + 29T^{2} \)
31 \( 1 + (2.44 - 4.23i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-9.69 + 5.59i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 1.46iT - 41T^{2} \)
43 \( 1 - 9.95T + 43T^{2} \)
47 \( 1 + (-5.30 + 3.06i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.03 + 2.32i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.55 + 6.16i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.19 - 1.26i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.0263 - 0.0456i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.212iT - 71T^{2} \)
73 \( 1 + (-7.43 + 12.8i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.399 - 0.230i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.9iT - 83T^{2} \)
89 \( 1 + (6.07 - 3.51i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 0.185T + 97T^{2} \)
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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

−10.53764919738803387288431796762, −9.584400655903988648186587951788, −8.531133992164955848602822345835, −7.62159417974508884694703711678, −6.73497348890372261878931692576, −6.13907019087926746625863759254, −5.44886037164685599224514295602, −4.56479923498339720794524765511, −2.28519460728039652710812137784, −0.53687160210253421575352785457, 0.68514306585376320189968664801, 2.74541968016351454133374620769, 4.16560738652823021347041127412, 4.64520098991671369582230835574, 5.78640755082195305105493754888, 6.94720939401191949737417110289, 7.81167706398649020387945560564, 9.359709516807034578981922251015, 9.841788488788571417293536812449, 10.45935036050292060910411530143

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