↑

Properties

Label 2-670-67.37-c1-0-14
Degree $2$
Conductor $670$
Sign $0.743 + 0.669i$
Analytic cond. $5.34997$
Root an. cond. $2.31300$
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  + (−0.5 − 0.866i)2-s + 2.41·3-s + (−0.499 + 0.866i)4-s + 5-s + (−1.20 − 2.09i)6-s + (0.707 − 1.22i)7-s + 0.999·8-s + 2.82·9-s + (−0.5 − 0.866i)10-s + (1 − 1.73i)11-s + (−1.20 + 2.09i)12-s + (3.27 + 5.67i)13-s − 1.41·14-s + 2.41·15-s + (−0.5 − 0.866i)16-s + (1.60 + 2.78i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + 1.39·3-s + (−0.249 + 0.433i)4-s + 0.447·5-s + (−0.492 − 0.853i)6-s + (0.267 − 0.462i)7-s + 0.353·8-s + 0.942·9-s + (−0.158 − 0.273i)10-s + (0.301 − 0.522i)11-s + (−0.348 + 0.603i)12-s + (0.908 + 1.57i)13-s − 0.377·14-s + 0.623·15-s + (−0.125 − 0.216i)16-s + (0.390 + 0.675i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.743 + 0.669i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.743 + 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(670\)    =    \(2 \cdot 5 \cdot 67\)
Sign: $0.743 + 0.669i$
Analytic conductor: \(5.34997\)
Root analytic conductor: \(2.31300\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{670} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 670,\ (\ :1/2),\ 0.743 + 0.669i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.04807 - 0.786201i\)
\(L(\frac12)\) \(\approx\) \(2.04807 - 0.786201i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 - T \)
67 \( 1 + (6.07 - 5.48i)T \)
good3 \( 1 - 2.41T + 3T^{2} \)
7 \( 1 + (-0.707 + 1.22i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.27 - 5.67i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.60 - 2.78i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.31 + 2.27i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.92 + 6.79i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.68 - 6.39i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.66 + 2.87i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.402 + 0.696i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.64 + 6.31i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 1.09T + 43T^{2} \)
47 \( 1 + (-1.15 + 1.99i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 0.918T + 53T^{2} \)
59 \( 1 + 4.63T + 59T^{2} \)
61 \( 1 + (-0.259 - 0.449i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (1.10 - 1.92i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.21 + 3.84i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.804 + 1.39i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.07 + 7.05i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 2.26T + 89T^{2} \)
97 \( 1 + (0.131 + 0.226i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.39725642512274554144893728746, −9.239524756086180951604873432719, −8.881468630366581396099728568414, −8.171913510824912295379352747002, −7.12130577630908543445978734780, −6.07283632304018494512621535282, −4.31510960535743524493465070814, −3.69765786707195159089718757070, −2.44688290845083838537990495505, −1.49813287139036922033154296350, 1.58166486892256921843443138852, 2.83817197207182245759929222007, 3.92722352845528630934290923625, 5.39457486898721394793230620085, 6.11729415399898732114275381965, 7.54868380948680025646122305034, 7.987986524357400718138094076382, 8.770185302255092050976947998441, 9.598706017276850237313356430819, 10.10656065746332939588820633353

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