↑

Properties

Label 2-700-140.19-c1-0-55
Degree $2$
Conductor $700$
Sign $0.936 + 0.350i$
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  + (−0.183 + 1.40i)2-s + (2.33 − 1.34i)3-s + (−1.93 − 0.514i)4-s + (1.45 + 3.51i)6-s + (0.0832 − 2.64i)7-s + (1.07 − 2.61i)8-s + (2.11 − 3.67i)9-s + (−1.31 + 0.760i)11-s + (−5.19 + 1.40i)12-s − 1.14·13-s + (3.69 + 0.601i)14-s + (3.47 + 1.98i)16-s + (2.21 + 3.83i)17-s + (4.76 + 3.64i)18-s + (3.04 − 5.28i)19-s + ⋯
L(s)  = 1  + (−0.129 + 0.991i)2-s + (1.34 − 0.776i)3-s + (−0.966 − 0.257i)4-s + (0.595 + 1.43i)6-s + (0.0314 − 0.999i)7-s + (0.380 − 0.924i)8-s + (0.706 − 1.22i)9-s + (−0.397 + 0.229i)11-s + (−1.49 + 0.404i)12-s − 0.316·13-s + (0.986 + 0.160i)14-s + (0.867 + 0.497i)16-s + (0.537 + 0.930i)17-s + (1.12 + 0.859i)18-s + (0.699 − 1.21i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.936 + 0.350i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ 0.936 + 0.350i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91680 - 0.347204i\)
\(L(\frac12)\) \(\approx\) \(1.91680 - 0.347204i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.183 - 1.40i)T \)
5 \( 1 \)
7 \( 1 + (-0.0832 + 2.64i)T \)
good3 \( 1 + (-2.33 + 1.34i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (1.31 - 0.760i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.14T + 13T^{2} \)
17 \( 1 + (-2.21 - 3.83i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.04 + 5.28i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.02 + 6.97i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.38T + 29T^{2} \)
31 \( 1 + (4.92 + 8.52i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-8.48 - 4.89i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.36iT - 41T^{2} \)
43 \( 1 + 4.89T + 43T^{2} \)
47 \( 1 + (-2.86 - 1.65i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.36 - 4.82i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.927 - 1.60i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.93 + 2.26i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.25 - 9.10i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.04iT - 71T^{2} \)
73 \( 1 + (-3.93 - 6.80i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.54 + 2.62i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.98iT - 83T^{2} \)
89 \( 1 + (-12.9 - 7.45i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 15.0T + 97T^{2} \)
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

−10.00744773634599658461930396290, −9.294401647749488112420746331042, −8.375604696418373497410014953738, −7.73474325175603658248739124793, −7.17139287438748424480372699567, −6.35480577856253037213724937647, −4.91425113974464613863031840971, −3.90739746478528755107358219739, −2.67928677739790121950300240126, −1.01072759232437139918950323324, 1.81645746282916077219776361621, 3.05814375481209681163978355413, 3.38649359399584135941405873024, 4.82288083547603806238634916015, 5.54172610219529042101013094365, 7.56742373319668392241975461228, 8.186313643881948381633970196075, 9.194571791137803719412834008410, 9.444923038698386883321814693181, 10.26254275821476688570886941379

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