↑

Properties

Label 2-930-465.119-c1-0-17
Degree $2$
Conductor $930$
Sign $0.907 - 0.420i$
Analytic cond. $7.42608$
Root an. cond. $2.72508$
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  − 2-s + (−1.67 + 0.454i)3-s + 4-s + (1.20 − 1.88i)5-s + (1.67 − 0.454i)6-s + (2.75 + 1.59i)7-s − 8-s + (2.58 − 1.51i)9-s + (−1.20 + 1.88i)10-s + (1.12 + 1.95i)11-s + (−1.67 + 0.454i)12-s + (1.41 + 2.45i)13-s + (−2.75 − 1.59i)14-s + (−1.15 + 3.69i)15-s + 16-s + (0.182 + 0.105i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.964 + 0.262i)3-s + 0.5·4-s + (0.537 − 0.843i)5-s + (0.682 − 0.185i)6-s + (1.04 + 0.601i)7-s − 0.353·8-s + (0.862 − 0.506i)9-s + (−0.379 + 0.596i)10-s + (0.340 + 0.589i)11-s + (−0.482 + 0.131i)12-s + (0.393 + 0.680i)13-s + (−0.736 − 0.425i)14-s + (−0.296 + 0.954i)15-s + 0.250·16-s + (0.0441 + 0.0255i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 - 0.420i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.907 - 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.907 - 0.420i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.907 - 0.420i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04529 + 0.230396i\)
\(L(\frac12)\) \(\approx\) \(1.04529 + 0.230396i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (1.67 - 0.454i)T \)
5 \( 1 + (-1.20 + 1.88i)T \)
31 \( 1 + (0.371 - 5.55i)T \)
good7 \( 1 + (-2.75 - 1.59i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.12 - 1.95i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.41 - 2.45i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.182 - 0.105i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.00830 + 0.0143i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 0.800iT - 23T^{2} \)
29 \( 1 + 0.452T + 29T^{2} \)
37 \( 1 + (-4.13 + 7.15i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.681 - 0.393i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.06 - 3.58i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 8.04T + 47T^{2} \)
53 \( 1 + (4.82 - 2.78i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.09 - 1.78i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 8.14iT - 61T^{2} \)
67 \( 1 + (-7.31 + 4.22i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (13.0 - 7.52i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.45 - 7.71i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-13.1 - 7.57i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-11.6 + 6.73i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 8.62T + 89T^{2} \)
97 \( 1 + 4.80iT - 97T^{2} \)
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

−10.07803150779065854368897605656, −9.226215034605140641558614725707, −8.738336275703908455922463611977, −7.67252603761169090277945193094, −6.64002392369292391524712878916, −5.76883951420462114936226074657, −4.98764398071027981814635483596, −4.14607293925944231449889795674, −2.07348541882393277688423343445, −1.16574767011857400136118689380, 0.906003542824016680491190825746, 2.05394353251956069637388388817, 3.57577562036733212444387056513, 4.94786229325308543856253701625, 5.93923377542351458221537279812, 6.56309855604695843449919489444, 7.55803595170903182958251290933, 8.070986531717543382475988208227, 9.350897456830615955782764303656, 10.26478062941618243501998863210

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