↑

Properties

Label 2-136-17.13-c3-0-5
Degree $2$
Conductor $136$
Sign $0.376 - 0.926i$
Analytic cond. $8.02425$
Root an. cond. $2.83271$
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  + (4.53 + 4.53i)3-s + (14.5 + 14.5i)5-s + (9.46 − 9.46i)7-s + 14.1i·9-s + (8.12 − 8.12i)11-s − 48.2·13-s + 132. i·15-s + (−19.1 − 67.4i)17-s + 33.5i·19-s + 85.8·21-s + (12.5 − 12.5i)23-s + 300. i·25-s + (58.1 − 58.1i)27-s + (−144. − 144. i)29-s + (125. + 125. i)31-s + ⋯
L(s)  = 1  + (0.873 + 0.873i)3-s + (1.30 + 1.30i)5-s + (0.510 − 0.510i)7-s + 0.525i·9-s + (0.222 − 0.222i)11-s − 1.02·13-s + 2.27i·15-s + (−0.273 − 0.961i)17-s + 0.404i·19-s + 0.892·21-s + (0.113 − 0.113i)23-s + 2.40i·25-s + (0.414 − 0.414i)27-s + (−0.927 − 0.927i)29-s + (0.727 + 0.727i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.376 - 0.926i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.376 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $0.376 - 0.926i$
Analytic conductor: \(8.02425\)
Root analytic conductor: \(2.83271\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :3/2),\ 0.376 - 0.926i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.19049 + 1.47460i\)
\(L(\frac12)\) \(\approx\) \(2.19049 + 1.47460i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 + (19.1 + 67.4i)T \)
good3 \( 1 + (-4.53 - 4.53i)T + 27iT^{2} \)
5 \( 1 + (-14.5 - 14.5i)T + 125iT^{2} \)
7 \( 1 + (-9.46 + 9.46i)T - 343iT^{2} \)
11 \( 1 + (-8.12 + 8.12i)T - 1.33e3iT^{2} \)
13 \( 1 + 48.2T + 2.19e3T^{2} \)
19 \( 1 - 33.5iT - 6.85e3T^{2} \)
23 \( 1 + (-12.5 + 12.5i)T - 1.21e4iT^{2} \)
29 \( 1 + (144. + 144. i)T + 2.43e4iT^{2} \)
31 \( 1 + (-125. - 125. i)T + 2.97e4iT^{2} \)
37 \( 1 + (18.9 + 18.9i)T + 5.06e4iT^{2} \)
41 \( 1 + (344. - 344. i)T - 6.89e4iT^{2} \)
43 \( 1 + 352. iT - 7.95e4T^{2} \)
47 \( 1 - 299.T + 1.03e5T^{2} \)
53 \( 1 + 347. iT - 1.48e5T^{2} \)
59 \( 1 + 511. iT - 2.05e5T^{2} \)
61 \( 1 + (-248. + 248. i)T - 2.26e5iT^{2} \)
67 \( 1 + 811.T + 3.00e5T^{2} \)
71 \( 1 + (107. + 107. i)T + 3.57e5iT^{2} \)
73 \( 1 + (653. + 653. i)T + 3.89e5iT^{2} \)
79 \( 1 + (-588. + 588. i)T - 4.93e5iT^{2} \)
83 \( 1 - 1.23e3iT - 5.71e5T^{2} \)
89 \( 1 - 428.T + 7.04e5T^{2} \)
97 \( 1 + (-373. - 373. i)T + 9.12e5iT^{2} \)
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

−13.46007962649551302046252892386, −11.67637628412348726358443236312, −10.48010480539287102859753765378, −9.915477710188823458088467274148, −9.082107902981820669217227262210, −7.56375302869771241864670998226, −6.44524576098281589299665277920, −4.92743844338140059089771208106, −3.36880307373346307331405202463, −2.25159096576282137173435757605, 1.49878591575231416342407811342, 2.35615342941138972975326862534, 4.74041156798443225874540414622, 5.83079489557074302739540771382, 7.30972310380277065171161491355, 8.561826131855879097994493253948, 9.052580395284638266534990015937, 10.21474922593306386816056602218, 11.97415017326566483835948235213, 12.81830844822129577357378658554

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