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Properties

Label 2-2e4-16.3-c8-0-7
Degree $2$
Conductor $16$
Sign $-0.357 - 0.934i$
Analytic cond. $6.51805$
Root an. cond. $2.55304$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (11.2 + 11.3i)2-s + (23.8 − 23.8i)3-s + (−3.51 + 255. i)4-s + (−695. + 695. i)5-s + (540. + 3.71i)6-s + 3.16e3·7-s + (−2.95e3 + 2.83e3i)8-s + 5.41e3i·9-s + (−1.57e4 − 108. i)10-s + (−295. − 295. i)11-s + (6.03e3 + 6.20e3i)12-s + (−2.04e4 − 2.04e4i)13-s + (3.55e4 + 3.60e4i)14-s + 3.32e4i·15-s + (−6.55e4 − 1.80e3i)16-s + 1.43e5·17-s + ⋯
L(s)  = 1  + (0.702 + 0.711i)2-s + (0.295 − 0.295i)3-s + (−0.0137 + 0.999i)4-s + (−1.11 + 1.11i)5-s + (0.417 + 0.00286i)6-s + 1.31·7-s + (−0.721 + 0.692i)8-s + 0.825i·9-s + (−1.57 − 0.0108i)10-s + (−0.0201 − 0.0201i)11-s + (0.290 + 0.299i)12-s + (−0.715 − 0.715i)13-s + (0.926 + 0.939i)14-s + 0.656i·15-s + (−0.999 − 0.0274i)16-s + 1.71·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.357 - 0.934i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.357 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.357 - 0.934i$
Analytic conductor: \(6.51805\)
Root analytic conductor: \(2.55304\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :4),\ -0.357 - 0.934i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.25931 + 1.82974i\)
\(L(\frac12)\) \(\approx\) \(1.25931 + 1.82974i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-11.2 - 11.3i)T \)
good3 \( 1 + (-23.8 + 23.8i)T - 6.56e3iT^{2} \)
5 \( 1 + (695. - 695. i)T - 3.90e5iT^{2} \)
7 \( 1 - 3.16e3T + 5.76e6T^{2} \)
11 \( 1 + (295. + 295. i)T + 2.14e8iT^{2} \)
13 \( 1 + (2.04e4 + 2.04e4i)T + 8.15e8iT^{2} \)
17 \( 1 - 1.43e5T + 6.97e9T^{2} \)
19 \( 1 + (-1.28e5 + 1.28e5i)T - 1.69e10iT^{2} \)
23 \( 1 + 6.18e4T + 7.83e10T^{2} \)
29 \( 1 + (-2.33e5 - 2.33e5i)T + 5.00e11iT^{2} \)
31 \( 1 - 3.74e5iT - 8.52e11T^{2} \)
37 \( 1 + (-5.16e5 + 5.16e5i)T - 3.51e12iT^{2} \)
41 \( 1 + 1.82e6iT - 7.98e12T^{2} \)
43 \( 1 + (8.34e5 + 8.34e5i)T + 1.16e13iT^{2} \)
47 \( 1 - 8.50e6iT - 2.38e13T^{2} \)
53 \( 1 + (3.16e6 - 3.16e6i)T - 6.22e13iT^{2} \)
59 \( 1 + (9.18e6 + 9.18e6i)T + 1.46e14iT^{2} \)
61 \( 1 + (1.32e5 + 1.32e5i)T + 1.91e14iT^{2} \)
67 \( 1 + (-1.64e5 + 1.64e5i)T - 4.06e14iT^{2} \)
71 \( 1 - 3.54e7T + 6.45e14T^{2} \)
73 \( 1 - 9.75e6iT - 8.06e14T^{2} \)
79 \( 1 + 4.97e7iT - 1.51e15T^{2} \)
83 \( 1 + (-5.00e7 + 5.00e7i)T - 2.25e15iT^{2} \)
89 \( 1 + 3.66e7iT - 3.93e15T^{2} \)
97 \( 1 - 1.20e8T + 7.83e15T^{2} \)
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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

−17.56606205699626312981992633351, −15.92829563501444580990079835665, −14.73240914942223632717468723204, −14.07721312474670391430704694309, −12.17214030834766047035394802360, −10.98105366150133188765627598787, −7.903243864961486705472169307102, −7.46703825402520554882036427125, −5.02083261264520933632519175546, −3.02029269234388302895288872005, 1.12256400143633796685932272583, 3.82632549944186383396072354354, 5.08061582704281342122675744497, 8.029832558865467833168458917680, 9.675104536530799532634873710552, 11.76602304068789072338914240456, 12.17547708535561446759770134379, 14.21709012452129560198512255424, 15.10873754181688792477704570154, 16.56605680724746028952590661130

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