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Properties

Label 2-384-48.35-c1-0-0
Degree $2$
Conductor $384$
Sign $-0.283 - 0.958i$
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  + (−1.43 − 0.966i)3-s + (−1.57 − 1.57i)5-s − 2.24·7-s + (1.13 + 2.77i)9-s + (−1.13 + 1.13i)11-s + (3.24 + 3.24i)13-s + (0.739 + 3.77i)15-s + 1.66i·17-s + (−3.77 + 3.77i)19-s + (3.23 + 2.17i)21-s + 2.26i·23-s − 0.0586i·25-s + (1.05 − 5.08i)27-s + (−3.23 + 3.23i)29-s − 1.30i·31-s + ⋯
L(s)  = 1  + (−0.829 − 0.558i)3-s + (−0.702 − 0.702i)5-s − 0.850·7-s + (0.377 + 0.926i)9-s + (−0.341 + 0.341i)11-s + (0.901 + 0.901i)13-s + (0.191 + 0.975i)15-s + 0.403i·17-s + (−0.866 + 0.866i)19-s + (0.705 + 0.474i)21-s + 0.471i·23-s − 0.0117i·25-s + (0.203 − 0.978i)27-s + (−0.600 + 0.600i)29-s − 0.234i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.283 - 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.283 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.283 - 0.958i$
Analytic conductor: \(3.06625\)
Root analytic conductor: \(1.75107\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :1/2),\ -0.283 - 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.172093 + 0.230341i\)
\(L(\frac12)\) \(\approx\) \(0.172093 + 0.230341i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.43 + 0.966i)T \)
good5 \( 1 + (1.57 + 1.57i)T + 5iT^{2} \)
7 \( 1 + 2.24T + 7T^{2} \)
11 \( 1 + (1.13 - 1.13i)T - 11iT^{2} \)
13 \( 1 + (-3.24 - 3.24i)T + 13iT^{2} \)
17 \( 1 - 1.66iT - 17T^{2} \)
19 \( 1 + (3.77 - 3.77i)T - 19iT^{2} \)
23 \( 1 - 2.26iT - 23T^{2} \)
29 \( 1 + (3.23 - 3.23i)T - 29iT^{2} \)
31 \( 1 + 1.30iT - 31T^{2} \)
37 \( 1 + (2.30 - 2.30i)T - 37iT^{2} \)
41 \( 1 + 10.2T + 41T^{2} \)
43 \( 1 + (-3.77 - 3.77i)T + 43iT^{2} \)
47 \( 1 + 3.74T + 47T^{2} \)
53 \( 1 + (-0.972 - 0.972i)T + 53iT^{2} \)
59 \( 1 + (3.88 - 3.88i)T - 59iT^{2} \)
61 \( 1 + (4.19 + 4.19i)T + 61iT^{2} \)
67 \( 1 + (-8.02 + 8.02i)T - 67iT^{2} \)
71 \( 1 - 11.0iT - 71T^{2} \)
73 \( 1 + 6.38iT - 73T^{2} \)
79 \( 1 + 2.69iT - 79T^{2} \)
83 \( 1 + (2.61 + 2.61i)T + 83iT^{2} \)
89 \( 1 - 7.35T + 89T^{2} \)
97 \( 1 + 5.67T + 97T^{2} \)
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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

−11.72824466466459221045243612996, −10.86040030461556821097253834806, −9.900020546514072145049606959924, −8.703782493052415225360099074856, −7.86612137397604012859456846903, −6.74782792816560352135899821074, −6.00411320264325196141772677939, −4.76358967739652170762560858752, −3.70901875720720953550199110328, −1.65713125531005149392823771458, 0.21147990622370046198353163745, 3.06266156293287945877871688570, 3.90705611129112182198485775409, 5.26751574405096642028511327174, 6.31081958381399970988409397188, 7.04339058385656194023696728216, 8.319508593930499594887705458702, 9.404887673072041444955278223150, 10.48394685429427027841732022169, 10.91665437358787336774772485118

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