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Properties

Label 2-384-32.29-c1-0-1
Degree $2$
Conductor $384$
Sign $0.233 - 0.972i$
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  + (−0.923 − 0.382i)3-s + (0.750 + 1.81i)5-s + (−0.638 + 0.638i)7-s + (0.707 + 0.707i)9-s + (0.343 − 0.142i)11-s + (−1.56 + 3.78i)13-s − 1.96i·15-s − 1.52i·17-s + (−3.15 + 7.61i)19-s + (0.834 − 0.345i)21-s + (6.00 + 6.00i)23-s + (0.813 − 0.813i)25-s + (−0.382 − 0.923i)27-s + (−0.647 − 0.268i)29-s + 3.66·31-s + ⋯
L(s)  = 1  + (−0.533 − 0.220i)3-s + (0.335 + 0.810i)5-s + (−0.241 + 0.241i)7-s + (0.235 + 0.235i)9-s + (0.103 − 0.0428i)11-s + (−0.435 + 1.05i)13-s − 0.506i·15-s − 0.369i·17-s + (−0.723 + 1.74i)19-s + (0.182 − 0.0754i)21-s + (1.25 + 1.25i)23-s + (0.162 − 0.162i)25-s + (−0.0736 − 0.177i)27-s + (−0.120 − 0.0497i)29-s + 0.658·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.233 - 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.233 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.233 - 0.972i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ 0.233 - 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.820155 + 0.646561i\)
\(L(\frac12)\) \(\approx\) \(0.820155 + 0.646561i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.923 + 0.382i)T \)
good5 \( 1 + (-0.750 - 1.81i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (0.638 - 0.638i)T - 7iT^{2} \)
11 \( 1 + (-0.343 + 0.142i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (1.56 - 3.78i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + 1.52iT - 17T^{2} \)
19 \( 1 + (3.15 - 7.61i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-6.00 - 6.00i)T + 23iT^{2} \)
29 \( 1 + (0.647 + 0.268i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 - 3.66T + 31T^{2} \)
37 \( 1 + (-3.69 - 8.90i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (8.19 + 8.19i)T + 41iT^{2} \)
43 \( 1 + (-1.86 + 0.771i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 3.21iT - 47T^{2} \)
53 \( 1 + (-7.71 + 3.19i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (2.78 + 6.72i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (10.4 + 4.34i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + (6.56 + 2.72i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (0.957 - 0.957i)T - 71iT^{2} \)
73 \( 1 + (2.14 + 2.14i)T + 73iT^{2} \)
79 \( 1 + 0.628iT - 79T^{2} \)
83 \( 1 + (-4.17 + 10.0i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (-8.70 + 8.70i)T - 89iT^{2} \)
97 \( 1 - 10.2T + 97T^{2} \)
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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

−11.63048729358575090625949281950, −10.58805529628253941659045031025, −9.892872622891064260627278983149, −8.877967589009085079768822346460, −7.57312068028180559324770478776, −6.68141459848382970896503241920, −5.97793706197007174881906386895, −4.74414959367361809228140560785, −3.30152308409423432690918015391, −1.85582635976311860682858585018, 0.76453973246980050044023110901, 2.77989550597439048016402733386, 4.45585944419597698070287018160, 5.15108879302238668978626426397, 6.28943591672129322945381200687, 7.26737965893873379982909604011, 8.593590114543772615437950609544, 9.273512546907104775483983370661, 10.37517936720271958670578059436, 10.97274832443217308629765817243

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