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Properties

Label 2-384-32.13-c1-0-1
Degree $2$
Conductor $384$
Sign $-0.648 - 0.761i$
Analytic cond. $3.06625$
Root an. cond. $1.75107$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.382 − 0.923i)3-s + (−2.51 + 1.04i)5-s + (−2.01 + 2.01i)7-s + (−0.707 − 0.707i)9-s + (1.32 + 3.20i)11-s + (−5.55 − 2.30i)13-s + 2.72i·15-s + 4.16i·17-s + (−4.49 − 1.86i)19-s + (1.08 + 2.62i)21-s + (1.94 + 1.94i)23-s + (1.69 − 1.69i)25-s + (−0.923 + 0.382i)27-s + (−1.96 + 4.73i)29-s + 2.87·31-s + ⋯
L(s)  = 1  + (0.220 − 0.533i)3-s + (−1.12 + 0.465i)5-s + (−0.759 + 0.759i)7-s + (−0.235 − 0.235i)9-s + (0.400 + 0.966i)11-s + (−1.54 − 0.638i)13-s + 0.702i·15-s + 1.01i·17-s + (−1.03 − 0.427i)19-s + (0.237 + 0.573i)21-s + (0.405 + 0.405i)23-s + (0.339 − 0.339i)25-s + (−0.177 + 0.0736i)27-s + (−0.364 + 0.879i)29-s + 0.515·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.648 - 0.761i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.648 - 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.648 - 0.761i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ -0.648 - 0.761i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.191678 + 0.415097i\)
\(L(\frac12)\) \(\approx\) \(0.191678 + 0.415097i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.382 + 0.923i)T \)
good5 \( 1 + (2.51 - 1.04i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (2.01 - 2.01i)T - 7iT^{2} \)
11 \( 1 + (-1.32 - 3.20i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (5.55 + 2.30i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 - 4.16iT - 17T^{2} \)
19 \( 1 + (4.49 + 1.86i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-1.94 - 1.94i)T + 23iT^{2} \)
29 \( 1 + (1.96 - 4.73i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 - 2.87T + 31T^{2} \)
37 \( 1 + (-6.31 + 2.61i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-0.756 - 0.756i)T + 41iT^{2} \)
43 \( 1 + (1.53 + 3.71i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 1.08iT - 47T^{2} \)
53 \( 1 + (2.90 + 7.01i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (10.3 - 4.28i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (2.97 - 7.18i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (-3.88 + 9.38i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-1.88 + 1.88i)T - 71iT^{2} \)
73 \( 1 + (7.00 + 7.00i)T + 73iT^{2} \)
79 \( 1 - 11.0iT - 79T^{2} \)
83 \( 1 + (-2.30 - 0.954i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (-7.65 + 7.65i)T - 89iT^{2} \)
97 \( 1 + 11.3T + 97T^{2} \)
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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

−11.92197640746663880294910477144, −10.83041160795772883565547835395, −9.785845116775692131961641243147, −8.862561557430225858465718788546, −7.75779285621955142202518192354, −7.12589832501133204697633324260, −6.16042145308636218485240610958, −4.70239601047215521970307146649, −3.41537690828936808312527225692, −2.30313621406796707754234662400, 0.28044110802369434133639258138, 2.88656903878539401618010777511, 4.07508142946631454873042348331, 4.69623006426651692206808962943, 6.31316108227839427568482748650, 7.37334598481966692422379150832, 8.223409799451713834123825373147, 9.267925689256164145773590762101, 9.965480938965999015475385953949, 11.09444089118932056821276362381

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