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Properties

Label 2-170-85.57-c1-0-3
Degree $2$
Conductor $170$
Sign $0.976 - 0.216i$
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  + (−0.382 − 0.923i)2-s + (−0.189 + 0.283i)3-s + (−0.707 + 0.707i)4-s + (1.79 + 1.32i)5-s + (0.334 + 0.0666i)6-s + (−0.701 + 3.52i)7-s + (0.923 + 0.382i)8-s + (1.10 + 2.66i)9-s + (0.540 − 2.16i)10-s + (0.940 − 4.72i)11-s + (−0.0666 − 0.334i)12-s − 1.94i·13-s + (3.52 − 0.701i)14-s + (−0.718 + 0.258i)15-s − i·16-s + (1.93 + 3.64i)17-s + ⋯
L(s)  = 1  + (−0.270 − 0.653i)2-s + (−0.109 + 0.163i)3-s + (−0.353 + 0.353i)4-s + (0.804 + 0.594i)5-s + (0.136 + 0.0271i)6-s + (−0.264 + 1.33i)7-s + (0.326 + 0.135i)8-s + (0.367 + 0.887i)9-s + (0.170 − 0.686i)10-s + (0.283 − 1.42i)11-s + (−0.0192 − 0.0966i)12-s − 0.538i·13-s + (0.941 − 0.187i)14-s + (−0.185 + 0.0666i)15-s − 0.250i·16-s + (0.468 + 0.883i)17-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

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.976 - 0.216i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.976 - 0.216i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05088 + 0.115274i\)
\(L(\frac12)\) \(\approx\) \(1.05088 + 0.115274i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.382 + 0.923i)T \)
5 \( 1 + (-1.79 - 1.32i)T \)
17 \( 1 + (-1.93 - 3.64i)T \)
good3 \( 1 + (0.189 - 0.283i)T + (-1.14 - 2.77i)T^{2} \)
7 \( 1 + (0.701 - 3.52i)T + (-6.46 - 2.67i)T^{2} \)
11 \( 1 + (-0.940 + 4.72i)T + (-10.1 - 4.20i)T^{2} \)
13 \( 1 + 1.94iT - 13T^{2} \)
19 \( 1 + (-0.750 + 1.81i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (1.30 - 0.870i)T + (8.80 - 21.2i)T^{2} \)
29 \( 1 + (-1.38 + 2.06i)T + (-11.0 - 26.7i)T^{2} \)
31 \( 1 + (1.47 + 7.43i)T + (-28.6 + 11.8i)T^{2} \)
37 \( 1 + (9.32 + 6.22i)T + (14.1 + 34.1i)T^{2} \)
41 \( 1 + (-2.81 - 4.20i)T + (-15.6 + 37.8i)T^{2} \)
43 \( 1 + (1.90 - 4.60i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + 5.22T + 47T^{2} \)
53 \( 1 + (-6.51 + 2.69i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-0.333 + 0.138i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-4.07 + 2.72i)T + (23.3 - 56.3i)T^{2} \)
67 \( 1 + (-8.03 + 8.03i)T - 67iT^{2} \)
71 \( 1 + (-7.79 + 1.54i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (3.19 + 16.0i)T + (-67.4 + 27.9i)T^{2} \)
79 \( 1 + (16.8 + 3.34i)T + (72.9 + 30.2i)T^{2} \)
83 \( 1 + (-1.13 - 2.74i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-9.67 + 9.67i)T - 89iT^{2} \)
97 \( 1 + (-1.24 - 6.26i)T + (-89.6 + 37.1i)T^{2} \)
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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

−12.83240819620084303402242882063, −11.60144542537565204450958329177, −10.80330109144958828093977448801, −9.916331706657368328753875830600, −8.945435419261279564053446096325, −7.927330657085038154008099201771, −6.16414598537608559351747783608, −5.37746117588323130298315188883, −3.36591525316520460297074542186, −2.14008188539804484405924630620, 1.32818921265181370590130651550, 4.03499802845743405195268873742, 5.20249075679800211366327018708, 6.80541433988191625623914553054, 7.10213463761212312185117257347, 8.744878254311385412246709426798, 9.829176411403294324119453827278, 10.17360930036263711368155906359, 12.01742644059468545479437248069, 12.79900967811055151799907923117

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