Properties

Label 2-700-35.4-c1-0-0
Degree $2$
Conductor $700$
Sign $-0.970 - 0.241i$
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  + (−0.315 − 0.182i)3-s + (−1.58 + 2.11i)7-s + (−1.43 − 2.48i)9-s + (−0.682 + 1.18i)11-s + 2.63i·13-s + (−1.93 − 1.11i)17-s + (−3.11 − 5.39i)19-s + (0.886 − 0.378i)21-s + (−5.71 + 3.29i)23-s + 2.13i·27-s − 5.50·29-s + (−2.25 + 3.89i)31-s + (0.430 − 0.248i)33-s + (−1.81 + 1.04i)37-s + (0.480 − 0.831i)39-s + ⋯
L(s)  = 1  + (−0.182 − 0.105i)3-s + (−0.600 + 0.799i)7-s + (−0.477 − 0.827i)9-s + (−0.205 + 0.356i)11-s + 0.730i·13-s + (−0.468 − 0.270i)17-s + (−0.714 − 1.23i)19-s + (0.193 − 0.0825i)21-s + (−1.19 + 0.687i)23-s + 0.411i·27-s − 1.02·29-s + (−0.404 + 0.700i)31-s + (0.0749 − 0.0432i)33-s + (−0.298 + 0.172i)37-s + (0.0769 − 0.133i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.970 - 0.241i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ -0.970 - 0.241i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0217411 + 0.177522i\)
\(L(\frac12)\) \(\approx\) \(0.0217411 + 0.177522i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 + (1.58 - 2.11i)T \)
good3 \( 1 + (0.315 + 0.182i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (0.682 - 1.18i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.63iT - 13T^{2} \)
17 \( 1 + (1.93 + 1.11i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.11 + 5.39i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.71 - 3.29i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.50T + 29T^{2} \)
31 \( 1 + (2.25 - 3.89i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.81 - 1.04i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 9.32T + 41T^{2} \)
43 \( 1 - 1.86iT - 43T^{2} \)
47 \( 1 + (5.94 - 3.43i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (8.78 + 5.06i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.817 - 1.41i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.0197 + 0.0341i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.82 - 3.36i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.27T + 71T^{2} \)
73 \( 1 + (3.46 + 2i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.66 - 4.61i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 14.7iT - 83T^{2} \)
89 \( 1 + (-0.433 - 0.750i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 16.0iT - 97T^{2} \)
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

−11.07893491341415392360759094439, −9.693089055154004211910916548266, −9.252310585305099143938674952144, −8.441502356112807758104655634151, −7.15693566926766723655190542941, −6.41045451659538567606393860013, −5.62012713016835507141202568214, −4.43681595905597864091841474099, −3.22447498408370107113307477000, −2.06450236633736994350945618332, 0.087911403545666118111851434217, 2.12144077633026346225763942704, 3.49085421330557671543741896192, 4.42164430829851612435987968996, 5.70725282292187072764518599280, 6.29742267737982035507134006512, 7.65750944448750227968620982838, 8.109427854158442001028310955872, 9.263046986472416216484969554539, 10.41585960744886553409958114182

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