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Properties

Label 2-2e4-16.13-c5-0-7
Degree $2$
Conductor $16$
Sign $-0.789 + 0.614i$
Analytic cond. $2.56614$
Root an. cond. $1.60191$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.379 + 5.64i)2-s + (−20.6 − 20.6i)3-s + (−31.7 + 4.28i)4-s + (−28.3 + 28.3i)5-s + (108. − 124. i)6-s − 55.5i·7-s + (−36.2 − 177. i)8-s + 609. i·9-s + (−170. − 149. i)10-s + (−137. + 137. i)11-s + (743. + 566. i)12-s + (−574. − 574. i)13-s + (313. − 21.0i)14-s + 1.16e3·15-s + (987. − 271. i)16-s + 320.·17-s + ⋯
L(s)  = 1  + (0.0671 + 0.997i)2-s + (−1.32 − 1.32i)3-s + (−0.990 + 0.133i)4-s + (−0.506 + 0.506i)5-s + (1.23 − 1.41i)6-s − 0.428i·7-s + (−0.200 − 0.979i)8-s + 2.50i·9-s + (−0.539 − 0.471i)10-s + (−0.342 + 0.342i)11-s + (1.48 + 1.13i)12-s + (−0.942 − 0.942i)13-s + (0.427 − 0.0287i)14-s + 1.34·15-s + (0.964 − 0.265i)16-s + 0.269·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.789 + 0.614i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.789 + 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.789 + 0.614i$
Analytic conductor: \(2.56614\)
Root analytic conductor: \(1.60191\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :5/2),\ -0.789 + 0.614i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0392581 - 0.114355i\)
\(L(\frac12)\) \(\approx\) \(0.0392581 - 0.114355i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.379 - 5.64i)T \)
good3 \( 1 + (20.6 + 20.6i)T + 243iT^{2} \)
5 \( 1 + (28.3 - 28.3i)T - 3.12e3iT^{2} \)
7 \( 1 + 55.5iT - 1.68e4T^{2} \)
11 \( 1 + (137. - 137. i)T - 1.61e5iT^{2} \)
13 \( 1 + (574. + 574. i)T + 3.71e5iT^{2} \)
17 \( 1 - 320.T + 1.41e6T^{2} \)
19 \( 1 + (858. + 858. i)T + 2.47e6iT^{2} \)
23 \( 1 + 825. iT - 6.43e6T^{2} \)
29 \( 1 + (-333. - 333. i)T + 2.05e7iT^{2} \)
31 \( 1 + 8.90e3T + 2.86e7T^{2} \)
37 \( 1 + (3.87e3 - 3.87e3i)T - 6.93e7iT^{2} \)
41 \( 1 + 6.54e3iT - 1.15e8T^{2} \)
43 \( 1 + (5.62e3 - 5.62e3i)T - 1.47e8iT^{2} \)
47 \( 1 + 3.12e3T + 2.29e8T^{2} \)
53 \( 1 + (-1.81e4 + 1.81e4i)T - 4.18e8iT^{2} \)
59 \( 1 + (5.53e3 - 5.53e3i)T - 7.14e8iT^{2} \)
61 \( 1 + (701. + 701. i)T + 8.44e8iT^{2} \)
67 \( 1 + (1.44e4 + 1.44e4i)T + 1.35e9iT^{2} \)
71 \( 1 + 2.36e4iT - 1.80e9T^{2} \)
73 \( 1 + 5.77e4iT - 2.07e9T^{2} \)
79 \( 1 + 5.78e4T + 3.07e9T^{2} \)
83 \( 1 + (-7.89e4 - 7.89e4i)T + 3.93e9iT^{2} \)
89 \( 1 - 2.59e4iT - 5.58e9T^{2} \)
97 \( 1 - 3.91e4T + 8.58e9T^{2} \)
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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

−17.51903179981328052097510362912, −16.58473769457496626968517778687, −14.98241725836097394494467109006, −13.29642534992990512947619069901, −12.30431069499245901605012380777, −10.62896991624589694633054934852, −7.70811720523750799451439408686, −6.90920566287481238749582251026, −5.27173199795994209428123613059, −0.11184407268388551157553779816, 4.13107145259953262085178448441, 5.42989290303281382266806998268, 9.074455794573723310149484051954, 10.34401874728858061143161799542, 11.59378641697458087288877766885, 12.39375020720885114303269852902, 14.74021172025697329372418588033, 16.21016417500276715246403851667, 17.16058893212688426841699678820, 18.59469512825189262584961550419

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