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Properties

Label 2-700-20.7-c1-0-12
Degree $2$
Conductor $700$
Sign $0.661 - 0.750i$
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  + (1.25 + 0.649i)2-s + (−2.28 − 2.28i)3-s + (1.15 + 1.63i)4-s + (−1.38 − 4.34i)6-s + (−0.707 + 0.707i)7-s + (0.393 + 2.80i)8-s + 7.41i·9-s + 1.40i·11-s + (1.08 − 6.36i)12-s + (0.699 − 0.699i)13-s + (−1.34 + 0.429i)14-s + (−1.32 + 3.77i)16-s + (3.26 + 3.26i)17-s + (−4.81 + 9.31i)18-s + 2.80·19-s + ⋯
L(s)  = 1  + (0.888 + 0.459i)2-s + (−1.31 − 1.31i)3-s + (0.578 + 0.815i)4-s + (−0.565 − 1.77i)6-s + (−0.267 + 0.267i)7-s + (0.139 + 0.990i)8-s + 2.47i·9-s + 0.424i·11-s + (0.312 − 1.83i)12-s + (0.193 − 0.193i)13-s + (−0.360 + 0.114i)14-s + (−0.331 + 0.943i)16-s + (0.792 + 0.792i)17-s + (−1.13 + 2.19i)18-s + 0.643·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 - 0.750i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.661 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.661 - 0.750i$
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: \((2,\ 700,\ (\ :1/2),\ 0.661 - 0.750i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40347 + 0.633854i\)
\(L(\frac12)\) \(\approx\) \(1.40347 + 0.633854i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.25 - 0.649i)T \)
5 \( 1 \)
7 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (2.28 + 2.28i)T + 3iT^{2} \)
11 \( 1 - 1.40iT - 11T^{2} \)
13 \( 1 + (-0.699 + 0.699i)T - 13iT^{2} \)
17 \( 1 + (-3.26 - 3.26i)T + 17iT^{2} \)
19 \( 1 - 2.80T + 19T^{2} \)
23 \( 1 + (-1.48 - 1.48i)T + 23iT^{2} \)
29 \( 1 + 4.85iT - 29T^{2} \)
31 \( 1 - 3.67iT - 31T^{2} \)
37 \( 1 + (-5.85 - 5.85i)T + 37iT^{2} \)
41 \( 1 - 3.53T + 41T^{2} \)
43 \( 1 + (-2.49 - 2.49i)T + 43iT^{2} \)
47 \( 1 + (1.86 - 1.86i)T - 47iT^{2} \)
53 \( 1 + (-0.696 + 0.696i)T - 53iT^{2} \)
59 \( 1 + 7.06T + 59T^{2} \)
61 \( 1 - 2.19T + 61T^{2} \)
67 \( 1 + (2.25 - 2.25i)T - 67iT^{2} \)
71 \( 1 - 12.1iT - 71T^{2} \)
73 \( 1 + (4.68 - 4.68i)T - 73iT^{2} \)
79 \( 1 + 0.599T + 79T^{2} \)
83 \( 1 + (4.53 + 4.53i)T + 83iT^{2} \)
89 \( 1 + 0.463iT - 89T^{2} \)
97 \( 1 + (8.00 + 8.00i)T + 97iT^{2} \)
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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

−11.04238744418712181387316113077, −9.946746956691291625882846606117, −8.305647182302173667972721651719, −7.59136368061827763845257373764, −6.85639983886459332772619193508, −5.99079335964076913237606373675, −5.53544322116491301190867317321, −4.44923334253439650631330905617, −2.85977898062870219301138668036, −1.45236311846987516205807163587, 0.78507001986627227252762104788, 3.09346860540562872373312495247, 3.95043228582514918366167775373, 4.84900510231182019960472981849, 5.59690992408135592961719621892, 6.28565565379927994091163835107, 7.33680062490023917843315197919, 9.227782288952553793744759814265, 9.730975649251686502644049166523, 10.67300148768139594840390863033

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