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Properties

Label 2-170-85.7-c1-0-2
Degree $2$
Conductor $170$
Sign $0.453 - 0.891i$
Analytic cond. $1.35745$
Root an. cond. $1.16509$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.923 − 0.382i)2-s + (0.572 + 2.87i)3-s + (0.707 − 0.707i)4-s + (−1.95 + 1.08i)5-s + (1.63 + 2.44i)6-s + (0.332 − 0.222i)7-s + (0.382 − 0.923i)8-s + (−5.18 + 2.14i)9-s + (−1.38 + 1.75i)10-s + (3.38 − 2.25i)11-s + (2.44 + 1.63i)12-s − 1.42i·13-s + (0.222 − 0.332i)14-s + (−4.25 − 5.00i)15-s − i·16-s + (3.58 + 2.04i)17-s + ⋯
L(s)  = 1  + (0.653 − 0.270i)2-s + (0.330 + 1.66i)3-s + (0.353 − 0.353i)4-s + (−0.873 + 0.486i)5-s + (0.665 + 0.996i)6-s + (0.125 − 0.0840i)7-s + (0.135 − 0.326i)8-s + (−1.72 + 0.716i)9-s + (−0.438 + 0.554i)10-s + (1.01 − 0.681i)11-s + (0.704 + 0.470i)12-s − 0.395i·13-s + (0.0594 − 0.0889i)14-s + (−1.09 − 1.29i)15-s − 0.250i·16-s + (0.868 + 0.495i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.453 - 0.891i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.453 - 0.891i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37241 + 0.841856i\)
\(L(\frac12)\) \(\approx\) \(1.37241 + 0.841856i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.923 + 0.382i)T \)
5 \( 1 + (1.95 - 1.08i)T \)
17 \( 1 + (-3.58 - 2.04i)T \)
good3 \( 1 + (-0.572 - 2.87i)T + (-2.77 + 1.14i)T^{2} \)
7 \( 1 + (-0.332 + 0.222i)T + (2.67 - 6.46i)T^{2} \)
11 \( 1 + (-3.38 + 2.25i)T + (4.20 - 10.1i)T^{2} \)
13 \( 1 + 1.42iT - 13T^{2} \)
19 \( 1 + (3.96 + 1.64i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-6.07 - 1.20i)T + (21.2 + 8.80i)T^{2} \)
29 \( 1 + (0.790 + 3.97i)T + (-26.7 + 11.0i)T^{2} \)
31 \( 1 + (5.97 + 3.99i)T + (11.8 + 28.6i)T^{2} \)
37 \( 1 + (-4.72 + 0.940i)T + (34.1 - 14.1i)T^{2} \)
41 \( 1 + (1.63 - 8.20i)T + (-37.8 - 15.6i)T^{2} \)
43 \( 1 + (10.7 + 4.46i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 + 7.15T + 47T^{2} \)
53 \( 1 + (1.83 + 4.41i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-1.95 - 4.72i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (10.8 + 2.14i)T + (56.3 + 23.3i)T^{2} \)
67 \( 1 + (-0.229 + 0.229i)T - 67iT^{2} \)
71 \( 1 + (-2.55 + 3.82i)T + (-27.1 - 65.5i)T^{2} \)
73 \( 1 + (-10.9 - 7.33i)T + (27.9 + 67.4i)T^{2} \)
79 \( 1 + (-0.586 - 0.877i)T + (-30.2 + 72.9i)T^{2} \)
83 \( 1 + (-1.14 + 0.476i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-9.95 + 9.95i)T - 89iT^{2} \)
97 \( 1 + (-2.47 - 1.65i)T + (37.1 + 89.6i)T^{2} \)
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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

−12.98481327290146644079156028102, −11.53408154795156074685027843003, −11.09390174109806395739279315876, −10.15713404913475357635785147710, −9.090786136073880740992220223010, −7.969437979182455811431932461274, −6.31614721398867839794882639880, −4.90934881338651827416736391834, −3.86212876496182692445212327485, −3.18061265175001255769855355570, 1.60178652820949691453539781023, 3.43083773318859475206691973130, 4.97650665613242446568330593776, 6.58092041831447232226892724598, 7.20661046450362462078152440475, 8.168196839689882689095758676860, 9.098221517152249784098757474808, 11.20416614997571663517064072441, 12.14333888133817510834593762043, 12.50964248889571591713610851801

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