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Properties

Label 2-2e4-4.3-c4-0-1
Degree $2$
Conductor $16$
Sign $0.5 + 0.866i$
Analytic cond. $1.65391$
Root an. cond. $1.28604$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 13.8i·3-s + 18·5-s + 27.7i·7-s − 110.·9-s + 124. i·11-s + 178·13-s − 249. i·15-s − 126·17-s − 401. i·19-s + 383.·21-s + 748. i·23-s − 301·25-s + 415. i·27-s − 1.42e3·29-s − 332. i·31-s + ⋯
L(s)  = 1  − 1.53i·3-s + 0.719·5-s + 0.565i·7-s − 1.37·9-s + 1.03i·11-s + 1.05·13-s − 1.10i·15-s − 0.435·17-s − 1.11i·19-s + 0.870·21-s + 1.41i·23-s − 0.481·25-s + 0.570i·27-s − 1.69·29-s − 0.346i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.5 + 0.866i$
Analytic conductor: \(1.65391\)
Root analytic conductor: \(1.28604\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :2),\ 0.5 + 0.866i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.09266 - 0.630851i\)
\(L(\frac12)\) \(\approx\) \(1.09266 - 0.630851i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + 13.8iT - 81T^{2} \)
5 \( 1 - 18T + 625T^{2} \)
7 \( 1 - 27.7iT - 2.40e3T^{2} \)
11 \( 1 - 124. iT - 1.46e4T^{2} \)
13 \( 1 - 178T + 2.85e4T^{2} \)
17 \( 1 + 126T + 8.35e4T^{2} \)
19 \( 1 + 401. iT - 1.30e5T^{2} \)
23 \( 1 - 748. iT - 2.79e5T^{2} \)
29 \( 1 + 1.42e3T + 7.07e5T^{2} \)
31 \( 1 + 332. iT - 9.23e5T^{2} \)
37 \( 1 - 530T + 1.87e6T^{2} \)
41 \( 1 - 162T + 2.82e6T^{2} \)
43 \( 1 + 1.53e3iT - 3.41e6T^{2} \)
47 \( 1 - 3.49e3iT - 4.87e6T^{2} \)
53 \( 1 - 594T + 7.89e6T^{2} \)
59 \( 1 + 2.36e3iT - 1.21e7T^{2} \)
61 \( 1 - 626T + 1.38e7T^{2} \)
67 \( 1 - 1.09e3iT - 2.01e7T^{2} \)
71 \( 1 + 7.73e3iT - 2.54e7T^{2} \)
73 \( 1 + 6.68e3T + 2.83e7T^{2} \)
79 \( 1 - 1.38e3iT - 3.89e7T^{2} \)
83 \( 1 + 4.61e3iT - 4.74e7T^{2} \)
89 \( 1 - 8.22e3T + 6.27e7T^{2} \)
97 \( 1 + 1.59e3T + 8.85e7T^{2} \)
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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

−18.06836023437643585814049186767, −17.44936760526088162295216829859, −15.36696591063106557177972043046, −13.64759522673011492851164922547, −12.89920524090815261321697102276, −11.46607753553729154749631859307, −9.217531861594741111994238977521, −7.41165828246141512397470417158, −5.94309129791928061554367690148, −1.92229369019212028723216941492, 3.86456321740832332010452948237, 5.83299655036539627994944390209, 8.715598346365770058092297180882, 10.12250259294611252202556742692, 11.08207130663272797544744924450, 13.46465136652306531144306691342, 14.67508805471552722410182275946, 16.15714647425047230237525206001, 16.84154656385198925609835861715, 18.49403857721936048837836211054

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