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: $\chi_{128} (17, \cdot )$
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

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

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