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Properties

Label 2-170-85.28-c1-0-0
Degree $2$
Conductor $170$
Sign $-0.868 - 0.496i$
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 + (−1.49 + 2.23i)3-s + (0.707 − 0.707i)4-s + (2.04 + 0.906i)5-s + (0.524 − 2.63i)6-s + (−0.501 + 2.52i)7-s + (−0.382 + 0.923i)8-s + (−1.61 − 3.89i)9-s + (−2.23 − 0.0552i)10-s + (−2.07 − 0.412i)11-s + (0.524 + 2.63i)12-s − 0.669·13-s + (−0.501 − 2.52i)14-s + (−5.07 + 3.21i)15-s − i·16-s + (−3.66 − 1.88i)17-s + ⋯
L(s)  = 1  + (−0.653 + 0.270i)2-s + (−0.861 + 1.28i)3-s + (0.353 − 0.353i)4-s + (0.914 + 0.405i)5-s + (0.213 − 1.07i)6-s + (−0.189 + 0.952i)7-s + (−0.135 + 0.326i)8-s + (−0.537 − 1.29i)9-s + (−0.706 − 0.0174i)10-s + (−0.625 − 0.124i)11-s + (0.151 + 0.760i)12-s − 0.185·13-s + (−0.133 − 0.673i)14-s + (−1.31 + 0.829i)15-s − 0.250i·16-s + (−0.889 − 0.455i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.868 - 0.496i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.868 - 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $-0.868 - 0.496i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ -0.868 - 0.496i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.161658 + 0.608516i\)
\(L(\frac12)\) \(\approx\) \(0.161658 + 0.608516i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.04 - 0.906i)T \)
17 \( 1 + (3.66 + 1.88i)T \)
good3 \( 1 + (1.49 - 2.23i)T + (-1.14 - 2.77i)T^{2} \)
7 \( 1 + (0.501 - 2.52i)T + (-6.46 - 2.67i)T^{2} \)
11 \( 1 + (2.07 + 0.412i)T + (10.1 + 4.20i)T^{2} \)
13 \( 1 + 0.669T + 13T^{2} \)
19 \( 1 + (2.43 - 5.87i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-4.02 + 2.68i)T + (8.80 - 21.2i)T^{2} \)
29 \( 1 + (-1.33 - 0.894i)T + (11.0 + 26.7i)T^{2} \)
31 \( 1 + (-8.76 + 1.74i)T + (28.6 - 11.8i)T^{2} \)
37 \( 1 + (-7.79 - 5.20i)T + (14.1 + 34.1i)T^{2} \)
41 \( 1 + (-1.26 + 0.844i)T + (15.6 - 37.8i)T^{2} \)
43 \( 1 + (6.08 + 2.52i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 8.92iT - 47T^{2} \)
53 \( 1 + (4.10 + 9.91i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (3.21 - 1.33i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-5.81 - 8.70i)T + (-23.3 + 56.3i)T^{2} \)
67 \( 1 + (-7.04 - 7.04i)T + 67iT^{2} \)
71 \( 1 + (1.31 + 6.58i)T + (-65.5 + 27.1i)T^{2} \)
73 \( 1 + (-2.39 - 12.0i)T + (-67.4 + 27.9i)T^{2} \)
79 \( 1 + (-1.27 + 6.40i)T + (-72.9 - 30.2i)T^{2} \)
83 \( 1 + (-11.2 + 4.65i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (0.324 - 0.324i)T - 89iT^{2} \)
97 \( 1 + (2.88 + 14.4i)T + (-89.6 + 37.1i)T^{2} \)
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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

−13.11928101894311169924565780949, −11.81268270511129065289876019941, −10.86576814026632243289288560059, −10.11041757644172805154060374219, −9.463000425179270046846721086313, −8.385596880374234148873633022011, −6.53023796904397430558264634461, −5.77719031285499686930771273408, −4.76411771153735880146697136100, −2.65613863118747924860600872589, 0.799756417731056049459499929176, 2.32015423322596195215318953721, 4.85659386013473989252280284510, 6.33919956542034656403410085802, 6.97653813188122305997421558339, 8.117989078357129341296788309340, 9.394635541011642936050460774013, 10.55042987353112696270454426320, 11.24358917676612024609888204763, 12.47308900674769356068923040417

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