↑

Properties

Label 2-700-20.7-c1-0-21
Degree $2$
Conductor $700$
Sign $-0.926 - 0.375i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.649 + 1.25i)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 + (−2.80 − 0.393i)8-s + 7.41i·9-s − 1.40i·11-s + (−6.36 + 1.08i)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 + (−9.31 + 4.81i)18-s − 2.80·19-s + ⋯
L(s)  = 1  + (0.459 + 0.888i)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 + (−0.990 − 0.139i)8-s + 2.47i·9-s − 0.424i·11-s + (−1.83 + 0.312i)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 + (−2.19 + 1.13i)18-s − 0.643·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 - 0.375i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.926 - 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.926 - 0.375i$
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: \((2,\ 700,\ (\ :1/2),\ -0.926 - 0.375i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.518698 + 2.66277i\)
\(L(\frac12)\) \(\approx\) \(0.518698 + 2.66277i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.649 - 1.25i)T \)
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 \( 1 + (8.00 + 8.00i)T + 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.49539518035482207276105126397, −9.802534096716400932239860853998, −8.906496335558631802738142345473, −8.152832126175343592035431929550, −7.77246947602414796293502265852, −6.27760642431856780987704845057, −5.23859420023409364894498017708, −4.21741784650719682388468834690, −3.69054732919950022975525247936, −2.56187231616002453417442669634, 1.19362431940318645660746240449, 2.21941808072961688372739330116, 3.06157112186947875718377359823, 4.13039353774646713296692806707, 5.52447668768779641162603078900, 6.63648585114469236718050946130, 7.53907260095178132826197274055, 8.473952697588687544194197336873, 9.166051117113056144629797220398, 9.918494243340447767884845173217

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