↑

Properties

Label 2-2e7-16.13-c3-0-5
Degree $2$
Conductor $128$
Sign $0.769 + 0.638i$
Analytic cond. $7.55224$
Root an. cond. $2.74813$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.98 + 1.98i)3-s + (0.596 − 0.596i)5-s − 29.0i·7-s − 19.1i·9-s + (12.1 − 12.1i)11-s + (48.5 + 48.5i)13-s + 2.36·15-s + 86.7·17-s + (−54.8 − 54.8i)19-s + (57.6 − 57.6i)21-s − 70.2i·23-s + 124. i·25-s + (91.5 − 91.5i)27-s + (−63.4 − 63.4i)29-s + 8.86·31-s + ⋯
L(s)  = 1  + (0.381 + 0.381i)3-s + (0.0533 − 0.0533i)5-s − 1.57i·7-s − 0.708i·9-s + (0.332 − 0.332i)11-s + (1.03 + 1.03i)13-s + 0.0407·15-s + 1.23·17-s + (−0.662 − 0.662i)19-s + (0.599 − 0.599i)21-s − 0.636i·23-s + 0.994i·25-s + (0.652 − 0.652i)27-s + (−0.405 − 0.405i)29-s + 0.0513·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.769 + 0.638i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.769 + 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.769 + 0.638i$
Analytic conductor: \(7.55224\)
Root analytic conductor: \(2.74813\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :3/2),\ 0.769 + 0.638i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.76857 - 0.638658i\)
\(L(\frac12)\) \(\approx\) \(1.76857 - 0.638658i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (-1.98 - 1.98i)T + 27iT^{2} \)
5 \( 1 + (-0.596 + 0.596i)T - 125iT^{2} \)
7 \( 1 + 29.0iT - 343T^{2} \)
11 \( 1 + (-12.1 + 12.1i)T - 1.33e3iT^{2} \)
13 \( 1 + (-48.5 - 48.5i)T + 2.19e3iT^{2} \)
17 \( 1 - 86.7T + 4.91e3T^{2} \)
19 \( 1 + (54.8 + 54.8i)T + 6.85e3iT^{2} \)
23 \( 1 + 70.2iT - 1.21e4T^{2} \)
29 \( 1 + (63.4 + 63.4i)T + 2.43e4iT^{2} \)
31 \( 1 - 8.86T + 2.97e4T^{2} \)
37 \( 1 + (-21.7 + 21.7i)T - 5.06e4iT^{2} \)
41 \( 1 + 153. iT - 6.89e4T^{2} \)
43 \( 1 + (120. - 120. i)T - 7.95e4iT^{2} \)
47 \( 1 - 99.9T + 1.03e5T^{2} \)
53 \( 1 + (389. - 389. i)T - 1.48e5iT^{2} \)
59 \( 1 + (324. - 324. i)T - 2.05e5iT^{2} \)
61 \( 1 + (-0.339 - 0.339i)T + 2.26e5iT^{2} \)
67 \( 1 + (-565. - 565. i)T + 3.00e5iT^{2} \)
71 \( 1 - 419. iT - 3.57e5T^{2} \)
73 \( 1 - 374. iT - 3.89e5T^{2} \)
79 \( 1 - 705.T + 4.93e5T^{2} \)
83 \( 1 + (947. + 947. i)T + 5.71e5iT^{2} \)
89 \( 1 + 4.72iT - 7.04e5T^{2} \)
97 \( 1 - 379.T + 9.12e5T^{2} \)
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

−12.94017859811864553688979504713, −11.56515224650387105624833961245, −10.64469064141941124670729581837, −9.593677649044582472644617713411, −8.641135026317235062405903001200, −7.27679321104616431113409427447, −6.23105277217611131807153981476, −4.30952400413741652789695886804, −3.50106726192023575850191302661, −1.05809930695375488544945376731, 1.82858430799991996478269439804, 3.23213485263434901861406194969, 5.25591428553470552524386161383, 6.21221886467040575430097505369, 7.903276746595869747324838003787, 8.506213165548772727868453785589, 9.771158643311466291879378882693, 10.96066386290738425975667889980, 12.20853525845220676382100137168, 12.82776654917252064787064890304

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