Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 17 $
Sign $-0.976 - 0.216i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·2-s − 3-s − 4-s + (−2 − i)5-s + i·6-s − 2·7-s + i·8-s − 2·9-s + (−1 + 2i)10-s + 12-s i·13-s + 2i·14-s + (2 + i)15-s + 16-s + (−1 − 4i)17-s + 2i·18-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577·3-s − 0.5·4-s + (−0.894 − 0.447i)5-s + 0.408i·6-s − 0.755·7-s + 0.353i·8-s − 0.666·9-s + (−0.316 + 0.632i)10-s + 0.288·12-s − 0.277i·13-s + 0.534i·14-s + (0.516 + 0.258i)15-s + 0.250·16-s + (−0.242 − 0.970i)17-s + 0.471i·18-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 - 0.216i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.976 - 0.216i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(170\)    =    \(2 \cdot 5 \cdot 17\)
\( \varepsilon \)  =  $-0.976 - 0.216i$
motivic weight  =  \(1\)
character  :  $\chi_{170} (169, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 170,\ (\ :1/2),\ -0.976 - 0.216i)$
$L(1)$  $\approx$  $0.0315332 + 0.287260i$
$L(\frac12)$  $\approx$  $0.0315332 + 0.287260i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;17\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;5,\;17\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + iT \)
5 \( 1 + (2 + i)T \)
17 \( 1 + (1 + 4i)T \)
good3 \( 1 + T + 3T^{2} \)
7 \( 1 + 2T + 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + iT - 13T^{2} \)
19 \( 1 + 5T + 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + 9iT - 29T^{2} \)
31 \( 1 - 5iT - 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + 10iT - 41T^{2} \)
43 \( 1 + 6iT - 43T^{2} \)
47 \( 1 - 7iT - 47T^{2} \)
53 \( 1 + iT - 53T^{2} \)
59 \( 1 + 5T + 59T^{2} \)
61 \( 1 - 5iT - 61T^{2} \)
67 \( 1 - 2iT - 67T^{2} \)
71 \( 1 - 5iT - 71T^{2} \)
73 \( 1 + 11T + 73T^{2} \)
79 \( 1 - 16iT - 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 - 5T + 89T^{2} \)
97 \( 1 + 7T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−12.15195208048770882825759031227, −11.35065363829323746908868029367, −10.52435747411357564780555327956, −9.225353027840115778268430985145, −8.365504559433248474709823801345, −6.93565938255421960397609114452, −5.53796867274967089762237965035, −4.31829051465682132741427742833, −2.92968787802752899023078845282, −0.27902629164556144119710329257, 3.29356522318068060642161603631, 4.67218221910896069750003341173, 6.15832968561240465992565053715, 6.79457098757508507053371387289, 8.102116035085813470266632876031, 9.027226062417982679320643995668, 10.48558930738402783262927176327, 11.25689181617923631173042783422, 12.39050079814818103875850422595, 13.19512286103300492115631524185

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