Properties

Degree $2$
Conductor $128$
Sign $0.768 - 0.640i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.92 + 7.06i)3-s + (13.5 − 5.60i)5-s + (23.0 − 23.0i)7-s + (−22.2 − 22.2i)9-s + (7.41 + 17.8i)11-s + (14.8 + 6.13i)13-s + 111. i·15-s + 27.7i·17-s + (82.2 + 34.0i)19-s + (95.5 + 230. i)21-s + (−39.2 − 39.2i)23-s + (63.1 − 63.1i)25-s + (31.8 − 13.2i)27-s + (5.57 − 13.4i)29-s − 155.·31-s + ⋯
L(s)  = 1  + (−0.563 + 1.36i)3-s + (1.20 − 0.501i)5-s + (1.24 − 1.24i)7-s + (−0.825 − 0.825i)9-s + (0.203 + 0.490i)11-s + (0.316 + 0.130i)13-s + 1.92i·15-s + 0.396i·17-s + (0.993 + 0.411i)19-s + (0.992 + 2.39i)21-s + (−0.355 − 0.355i)23-s + (0.504 − 0.504i)25-s + (0.227 − 0.0941i)27-s + (0.0357 − 0.0861i)29-s − 0.902·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.768 - 0.640i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.768 - 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.768 - 0.640i$
Motivic weight: \(3\)
Character: $\chi_{128} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :3/2),\ 0.768 - 0.640i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.76458 + 0.638699i\)
\(L(\frac12)\) \(\approx\) \(1.76458 + 0.638699i\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.92 - 7.06i)T + (-19.0 - 19.0i)T^{2} \)
5 \( 1 + (-13.5 + 5.60i)T + (88.3 - 88.3i)T^{2} \)
7 \( 1 + (-23.0 + 23.0i)T - 343iT^{2} \)
11 \( 1 + (-7.41 - 17.8i)T + (-941. + 941. i)T^{2} \)
13 \( 1 + (-14.8 - 6.13i)T + (1.55e3 + 1.55e3i)T^{2} \)
17 \( 1 - 27.7iT - 4.91e3T^{2} \)
19 \( 1 + (-82.2 - 34.0i)T + (4.85e3 + 4.85e3i)T^{2} \)
23 \( 1 + (39.2 + 39.2i)T + 1.21e4iT^{2} \)
29 \( 1 + (-5.57 + 13.4i)T + (-1.72e4 - 1.72e4i)T^{2} \)
31 \( 1 + 155.T + 2.97e4T^{2} \)
37 \( 1 + (-188. + 78.1i)T + (3.58e4 - 3.58e4i)T^{2} \)
41 \( 1 + (-113. - 113. i)T + 6.89e4iT^{2} \)
43 \( 1 + (-24.6 - 59.5i)T + (-5.62e4 + 5.62e4i)T^{2} \)
47 \( 1 - 217. iT - 1.03e5T^{2} \)
53 \( 1 + (35.7 + 86.3i)T + (-1.05e5 + 1.05e5i)T^{2} \)
59 \( 1 + (-116. + 48.2i)T + (1.45e5 - 1.45e5i)T^{2} \)
61 \( 1 + (197. - 475. i)T + (-1.60e5 - 1.60e5i)T^{2} \)
67 \( 1 + (144. - 349. i)T + (-2.12e5 - 2.12e5i)T^{2} \)
71 \( 1 + (-523. + 523. i)T - 3.57e5iT^{2} \)
73 \( 1 + (718. + 718. i)T + 3.89e5iT^{2} \)
79 \( 1 + 958. iT - 4.93e5T^{2} \)
83 \( 1 + (1.24e3 + 514. i)T + (4.04e5 + 4.04e5i)T^{2} \)
89 \( 1 + (808. - 808. i)T - 7.04e5iT^{2} \)
97 \( 1 - 1.39e3T + 9.12e5T^{2} \)
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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

−13.12834698027832650614212129396, −11.62001834416803137485049136261, −10.69738327654127417800098542166, −10.01162357553601558798990206713, −9.136829349322903390559735967709, −7.64322740060202127705191540690, −5.91680725207892362703656012689, −4.90212142793598849766246036705, −4.05556637664272938483553649330, −1.45435400389015609143760846034, 1.41140106709282889693970558839, 2.48551088428091323079161133339, 5.41029350562365617106151593781, 5.94308741298637300763193621473, 7.16557170287037863367613254559, 8.343996842497851503098247904382, 9.525409050955842384076545988964, 11.12440704826187360884372317157, 11.68907229804964840821590268251, 12.73461069210294279834393842435

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