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Properties

Label 2-2e4-16.5-c9-0-10
Degree $2$
Conductor $16$
Sign $0.998 + 0.0562i$
Analytic cond. $8.24057$
Root an. cond. $2.87063$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (17.2 + 14.5i)2-s + (185. − 185. i)3-s + (85.7 + 504. i)4-s + (356. + 356. i)5-s + (5.90e3 − 497. i)6-s − 6.40e3i·7-s + (−5.88e3 + 9.97e3i)8-s − 4.87e4i·9-s + (957. + 1.13e4i)10-s + (5.00e4 + 5.00e4i)11-s + (1.09e5 + 7.75e4i)12-s + (−6.08e4 + 6.08e4i)13-s + (9.35e4 − 1.10e5i)14-s + 1.31e5·15-s + (−2.47e5 + 8.65e4i)16-s − 1.43e5·17-s + ⋯
L(s)  = 1  + (0.764 + 0.645i)2-s + (1.31 − 1.31i)3-s + (0.167 + 0.985i)4-s + (0.254 + 0.254i)5-s + (1.85 − 0.156i)6-s − 1.00i·7-s + (−0.508 + 0.861i)8-s − 2.47i·9-s + (0.0302 + 0.359i)10-s + (1.03 + 1.03i)11-s + (1.52 + 1.07i)12-s + (−0.590 + 0.590i)13-s + (0.650 − 0.770i)14-s + 0.672·15-s + (−0.943 + 0.330i)16-s − 0.417·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0562i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0562i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.998 + 0.0562i$
Analytic conductor: \(8.24057\)
Root analytic conductor: \(2.87063\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :9/2),\ 0.998 + 0.0562i)\)

Particular Values

\(L(5)\) \(\approx\) \(3.67963 - 0.103506i\)
\(L(\frac12)\) \(\approx\) \(3.67963 - 0.103506i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-17.2 - 14.5i)T \)
good3 \( 1 + (-185. + 185. i)T - 1.96e4iT^{2} \)
5 \( 1 + (-356. - 356. i)T + 1.95e6iT^{2} \)
7 \( 1 + 6.40e3iT - 4.03e7T^{2} \)
11 \( 1 + (-5.00e4 - 5.00e4i)T + 2.35e9iT^{2} \)
13 \( 1 + (6.08e4 - 6.08e4i)T - 1.06e10iT^{2} \)
17 \( 1 + 1.43e5T + 1.18e11T^{2} \)
19 \( 1 + (-3.43e4 + 3.43e4i)T - 3.22e11iT^{2} \)
23 \( 1 - 1.27e6iT - 1.80e12T^{2} \)
29 \( 1 + (3.36e6 - 3.36e6i)T - 1.45e13iT^{2} \)
31 \( 1 + 6.77e6T + 2.64e13T^{2} \)
37 \( 1 + (-5.33e6 - 5.33e6i)T + 1.29e14iT^{2} \)
41 \( 1 + 6.67e6iT - 3.27e14T^{2} \)
43 \( 1 + (1.55e7 + 1.55e7i)T + 5.02e14iT^{2} \)
47 \( 1 - 1.73e7T + 1.11e15T^{2} \)
53 \( 1 + (-3.02e7 - 3.02e7i)T + 3.29e15iT^{2} \)
59 \( 1 + (-9.62e7 - 9.62e7i)T + 8.66e15iT^{2} \)
61 \( 1 + (-8.23e7 + 8.23e7i)T - 1.16e16iT^{2} \)
67 \( 1 + (-3.79e7 + 3.79e7i)T - 2.72e16iT^{2} \)
71 \( 1 + 3.18e8iT - 4.58e16T^{2} \)
73 \( 1 + 1.21e7iT - 5.88e16T^{2} \)
79 \( 1 - 1.96e8T + 1.19e17T^{2} \)
83 \( 1 + (-1.10e8 + 1.10e8i)T - 1.86e17iT^{2} \)
89 \( 1 + 4.70e8iT - 3.50e17T^{2} \)
97 \( 1 - 1.68e8T + 7.60e17T^{2} \)
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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

−17.03286348952137475157752533891, −14.90688381790337550465407826198, −14.18058355462527081120577426608, −13.25942570540167450363949290820, −12.06169758451273666796293270557, −9.160150241810670775169804847123, −7.41909676969620301216595979709, −6.81065845637058161711894104357, −3.83126838099641933022256221469, −1.96280821414178630957666326981, 2.40831634548954494381956906951, 3.78884933866659935181714317913, 5.42460099502029479299128465873, 8.769710623719734686877042449819, 9.681104211417121007003977511895, 11.23847937137892887372237222676, 13.10566432363568866427209796748, 14.45625753120285521285080883531, 15.11398291401165288399645394913, 16.40182918211491515464489488724

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