Properties

Degree $2$
Conductor $128$
Sign $0.641 + 0.767i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.729 − 1.76i)3-s + (4.29 + 1.78i)5-s + (−1.47 − 1.47i)7-s + (16.5 − 16.5i)9-s + (−0.854 + 2.06i)11-s + (40.9 − 16.9i)13-s − 8.86i·15-s − 73.1i·17-s + (18.2 − 7.55i)19-s + (−1.52 + 3.68i)21-s + (144. − 144. i)23-s + (−73.0 − 73.0i)25-s + (−88.6 − 36.7i)27-s + (80.7 + 194. i)29-s + 168.·31-s + ⋯
L(s)  = 1  + (−0.140 − 0.338i)3-s + (0.384 + 0.159i)5-s + (−0.0798 − 0.0798i)7-s + (0.611 − 0.611i)9-s + (−0.0234 + 0.0565i)11-s + (0.874 − 0.362i)13-s − 0.152i·15-s − 1.04i·17-s + (0.220 − 0.0912i)19-s + (−0.0158 + 0.0382i)21-s + (1.30 − 1.30i)23-s + (−0.584 − 0.584i)25-s + (−0.632 − 0.261i)27-s + (0.517 + 1.24i)29-s + 0.978·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.641 + 0.767i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.641 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.641 + 0.767i$
Motivic weight: \(3\)
Character: $\chi_{128} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :3/2),\ 0.641 + 0.767i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.54722 - 0.723088i\)
\(L(\frac12)\) \(\approx\) \(1.54722 - 0.723088i\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.729 + 1.76i)T + (-19.0 + 19.0i)T^{2} \)
5 \( 1 + (-4.29 - 1.78i)T + (88.3 + 88.3i)T^{2} \)
7 \( 1 + (1.47 + 1.47i)T + 343iT^{2} \)
11 \( 1 + (0.854 - 2.06i)T + (-941. - 941. i)T^{2} \)
13 \( 1 + (-40.9 + 16.9i)T + (1.55e3 - 1.55e3i)T^{2} \)
17 \( 1 + 73.1iT - 4.91e3T^{2} \)
19 \( 1 + (-18.2 + 7.55i)T + (4.85e3 - 4.85e3i)T^{2} \)
23 \( 1 + (-144. + 144. i)T - 1.21e4iT^{2} \)
29 \( 1 + (-80.7 - 194. i)T + (-1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 - 168.T + 2.97e4T^{2} \)
37 \( 1 + (72.0 + 29.8i)T + (3.58e4 + 3.58e4i)T^{2} \)
41 \( 1 + (141. - 141. i)T - 6.89e4iT^{2} \)
43 \( 1 + (161. - 389. i)T + (-5.62e4 - 5.62e4i)T^{2} \)
47 \( 1 - 239. iT - 1.03e5T^{2} \)
53 \( 1 + (59.8 - 144. i)T + (-1.05e5 - 1.05e5i)T^{2} \)
59 \( 1 + (582. + 241. i)T + (1.45e5 + 1.45e5i)T^{2} \)
61 \( 1 + (-238. - 575. i)T + (-1.60e5 + 1.60e5i)T^{2} \)
67 \( 1 + (-156. - 376. i)T + (-2.12e5 + 2.12e5i)T^{2} \)
71 \( 1 + (411. + 411. i)T + 3.57e5iT^{2} \)
73 \( 1 + (642. - 642. i)T - 3.89e5iT^{2} \)
79 \( 1 - 800. iT - 4.93e5T^{2} \)
83 \( 1 + (-1.34e3 + 557. i)T + (4.04e5 - 4.04e5i)T^{2} \)
89 \( 1 + (-340. - 340. i)T + 7.04e5iT^{2} \)
97 \( 1 - 632.T + 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.79692620804087101397052452390, −11.78549159246711436507301525038, −10.62902790891765098492477175379, −9.636760079368364537858056536505, −8.488959562312689293190693765794, −7.06220950477101865263180272491, −6.24828234514340362475912445237, −4.71815363273205143108506137628, −3.01556575419237639116126540584, −1.04153292066156965486272647081, 1.65092959512903089315916627287, 3.68862335266222295642234708180, 5.07712174498136307876370111322, 6.25818534225989944220685076847, 7.65322177824216107767195475683, 8.889832891225005454321289088682, 9.952235708932004339440637143689, 10.84978187607420285528820790715, 11.91421832935309866647286067588, 13.29334976303844323556819471285

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