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Properties

Label 2-384-32.5-c1-0-4
Degree $2$
Conductor $384$
Sign $0.945 + 0.324i$
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 + (−1.46 − 0.605i)5-s + (3.54 + 3.54i)7-s + (−0.707 + 0.707i)9-s + (0.471 − 1.13i)11-s + (4.97 − 2.05i)13-s + 1.58i·15-s + 0.419i·17-s + (0.721 − 0.298i)19-s + (1.92 − 4.63i)21-s + (5.76 − 5.76i)23-s + (−1.76 − 1.76i)25-s + (0.923 + 0.382i)27-s + (1.26 + 3.05i)29-s − 0.702·31-s + ⋯
L(s)  = 1  + (−0.220 − 0.533i)3-s + (−0.653 − 0.270i)5-s + (1.34 + 1.34i)7-s + (−0.235 + 0.235i)9-s + (0.142 − 0.342i)11-s + (1.37 − 0.570i)13-s + 0.408i·15-s + 0.101i·17-s + (0.165 − 0.0685i)19-s + (0.419 − 1.01i)21-s + (1.20 − 1.20i)23-s + (−0.352 − 0.352i)25-s + (0.177 + 0.0736i)27-s + (0.234 + 0.566i)29-s − 0.126·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.324i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 + 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.945 + 0.324i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ 0.945 + 0.324i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34219 - 0.223986i\)
\(L(\frac12)\) \(\approx\) \(1.34219 - 0.223986i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.46 + 0.605i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (-3.54 - 3.54i)T + 7iT^{2} \)
11 \( 1 + (-0.471 + 1.13i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (-4.97 + 2.05i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 - 0.419iT - 17T^{2} \)
19 \( 1 + (-0.721 + 0.298i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-5.76 + 5.76i)T - 23iT^{2} \)
29 \( 1 + (-1.26 - 3.05i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + 0.702T + 31T^{2} \)
37 \( 1 + (-1.86 - 0.773i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-1.76 + 1.76i)T - 41iT^{2} \)
43 \( 1 + (1.70 - 4.12i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 - 9.64iT - 47T^{2} \)
53 \( 1 + (0.729 - 1.76i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (9.04 + 3.74i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (-0.0348 - 0.0842i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (1.84 + 4.44i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (-4.81 - 4.81i)T + 71iT^{2} \)
73 \( 1 + (-4.70 + 4.70i)T - 73iT^{2} \)
79 \( 1 + 2.83iT - 79T^{2} \)
83 \( 1 + (8.15 - 3.37i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (5.34 + 5.34i)T + 89iT^{2} \)
97 \( 1 + 10.5T + 97T^{2} \)
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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

−11.20591343703461737809745669315, −10.92347264116161001330291704773, −9.074994717040670247555570721751, −8.378306920279466829003423739671, −7.87694362569502901927443003103, −6.40285557683250622240380304111, −5.50635743315164987037732761272, −4.50247927162059594834612226705, −2.87014664995168823966760296278, −1.30155986417255048221681180875, 1.34823215804098064538529466404, 3.63221240790245439711764960436, 4.26448430897662493024260488851, 5.35730276302448146613026763073, 6.84611722386058410626479662493, 7.62698667278162082108561104619, 8.533762554293903214003822288843, 9.670401667249973598218950904024, 10.81285257749389448622727995480, 11.20070217468197281552271989762

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