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Properties

Label 2-384-32.13-c1-0-3
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: $\chi_{384} (145, \cdot )$
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

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

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