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Properties

Label 2-170-85.58-c1-0-0
Degree $2$
Conductor $170$
Sign $-0.939 + 0.343i$
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 + (−2.47 + 0.491i)3-s + (−0.707 + 0.707i)4-s + (0.0805 + 2.23i)5-s + (−1.39 − 2.09i)6-s + (−2.60 − 3.89i)7-s + (−0.923 − 0.382i)8-s + (3.09 − 1.28i)9-s + (−2.03 + 0.929i)10-s + (−4.03 + 2.69i)11-s + (1.39 − 2.09i)12-s + 0.696·13-s + (2.60 − 3.89i)14-s + (−1.29 − 5.48i)15-s − i·16-s + (−3.92 + 1.26i)17-s + ⋯
L(s)  = 1  + (0.270 + 0.653i)2-s + (−1.42 + 0.283i)3-s + (−0.353 + 0.353i)4-s + (0.0360 + 0.999i)5-s + (−0.571 − 0.855i)6-s + (−0.984 − 1.47i)7-s + (−0.326 − 0.135i)8-s + (1.03 − 0.426i)9-s + (−0.643 + 0.293i)10-s + (−1.21 + 0.812i)11-s + (0.404 − 0.604i)12-s + 0.193·13-s + (0.696 − 1.04i)14-s + (−0.334 − 1.41i)15-s − 0.250i·16-s + (−0.952 + 0.306i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.343i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $-0.939 + 0.343i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ -0.939 + 0.343i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0553953 - 0.312668i\)
\(L(\frac12)\) \(\approx\) \(0.0553953 - 0.312668i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0805 - 2.23i)T \)
17 \( 1 + (3.92 - 1.26i)T \)
good3 \( 1 + (2.47 - 0.491i)T + (2.77 - 1.14i)T^{2} \)
7 \( 1 + (2.60 + 3.89i)T + (-2.67 + 6.46i)T^{2} \)
11 \( 1 + (4.03 - 2.69i)T + (4.20 - 10.1i)T^{2} \)
13 \( 1 - 0.696T + 13T^{2} \)
19 \( 1 + (-3.16 - 1.31i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (1.09 - 5.49i)T + (-21.2 - 8.80i)T^{2} \)
29 \( 1 + (-1.47 - 7.40i)T + (-26.7 + 11.0i)T^{2} \)
31 \( 1 + (-1.62 - 1.08i)T + (11.8 + 28.6i)T^{2} \)
37 \( 1 + (0.403 + 2.02i)T + (-34.1 + 14.1i)T^{2} \)
41 \( 1 + (-1.40 + 7.04i)T + (-37.8 - 15.6i)T^{2} \)
43 \( 1 + (1.74 - 4.22i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 - 2.85iT - 47T^{2} \)
53 \( 1 + (1.13 - 0.469i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (3.91 + 9.44i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (4.67 + 0.930i)T + (56.3 + 23.3i)T^{2} \)
67 \( 1 + (-0.385 - 0.385i)T + 67iT^{2} \)
71 \( 1 + (0.365 - 0.546i)T + (-27.1 - 65.5i)T^{2} \)
73 \( 1 + (-0.645 + 0.966i)T + (-27.9 - 67.4i)T^{2} \)
79 \( 1 + (1.73 + 2.60i)T + (-30.2 + 72.9i)T^{2} \)
83 \( 1 + (-3.48 - 8.40i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-6.89 + 6.89i)T - 89iT^{2} \)
97 \( 1 + (-0.945 + 1.41i)T + (-37.1 - 89.6i)T^{2} \)
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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

−13.32694791321295032756466344493, −12.44873263648197057703652092425, −11.08961823741380674869889498020, −10.47410300853278726174930585164, −9.731895688257746606938305661454, −7.54999534407938106901616183682, −6.91381019371045670255426672157, −6.01668736569735496896892226629, −4.79621333981683874101517789930, −3.49258743983735080856262240762, 0.30600204218951802258379159308, 2.64555038030248853943173594535, 4.76822877386703622103601853107, 5.65671028207759290200765040812, 6.30713596444040171654479711025, 8.344488493421646374464684894768, 9.335831965239328151224166918014, 10.45847546677156375446828611570, 11.62075218171842770071182620214, 12.06839618608017643887099328281

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