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Properties

Label 2-170-85.37-c1-0-2
Degree $2$
Conductor $170$
Sign $0.701 - 0.712i$
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 + (−0.283 + 0.189i)3-s + (0.707 + 0.707i)4-s + (−1.91 + 1.15i)5-s + (0.334 − 0.0666i)6-s + (3.52 − 0.701i)7-s + (−0.382 − 0.923i)8-s + (−1.10 + 2.66i)9-s + (2.21 − 0.331i)10-s + (0.940 + 4.72i)11-s + (−0.334 − 0.0666i)12-s + 1.94·13-s + (−3.52 − 0.701i)14-s + (0.325 − 0.690i)15-s + i·16-s + (3.64 + 1.93i)17-s + ⋯
L(s)  = 1  + (−0.653 − 0.270i)2-s + (−0.163 + 0.109i)3-s + (0.353 + 0.353i)4-s + (−0.856 + 0.515i)5-s + (0.136 − 0.0271i)6-s + (1.33 − 0.264i)7-s + (−0.135 − 0.326i)8-s + (−0.367 + 0.887i)9-s + (0.699 − 0.104i)10-s + (0.283 + 1.42i)11-s + (−0.0966 − 0.0192i)12-s + 0.538·13-s + (−0.941 − 0.187i)14-s + (0.0840 − 0.178i)15-s + 0.250i·16-s + (0.883 + 0.468i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.701 - 0.712i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.701 - 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.723282 + 0.302978i\)
\(L(\frac12)\) \(\approx\) \(0.723282 + 0.302978i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.91 - 1.15i)T \)
17 \( 1 + (-3.64 - 1.93i)T \)
good3 \( 1 + (0.283 - 0.189i)T + (1.14 - 2.77i)T^{2} \)
7 \( 1 + (-3.52 + 0.701i)T + (6.46 - 2.67i)T^{2} \)
11 \( 1 + (-0.940 - 4.72i)T + (-10.1 + 4.20i)T^{2} \)
13 \( 1 - 1.94T + 13T^{2} \)
19 \( 1 + (0.750 + 1.81i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (0.870 - 1.30i)T + (-8.80 - 21.2i)T^{2} \)
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 \( 1 + (6.22 + 9.32i)T + (-14.1 + 34.1i)T^{2} \)
41 \( 1 + (-2.81 + 4.20i)T + (-15.6 - 37.8i)T^{2} \)
43 \( 1 + (4.60 - 1.90i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 5.22iT - 47T^{2} \)
53 \( 1 + (-2.69 + 6.51i)T + (-37.4 - 37.4i)T^{2} \)
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 \( 1 + (-16.0 - 3.19i)T + (67.4 + 27.9i)T^{2} \)
79 \( 1 + (-16.8 + 3.34i)T + (72.9 - 30.2i)T^{2} \)
83 \( 1 + (2.74 + 1.13i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (9.67 + 9.67i)T + 89iT^{2} \)
97 \( 1 + (-6.26 - 1.24i)T + (89.6 + 37.1i)T^{2} \)
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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

−12.51781575222040431057176110244, −11.63190106189311196626602290583, −10.89635991805162600750013469690, −10.20019779470141856854239961482, −8.636898752193427482109608221920, −7.80131945294864273868086725328, −7.05334200495658405069373534494, −5.14249650111416390259154363282, −3.88536729180681933653115851364, −1.94161725938200969652879556106, 1.04545082077145464496867663400, 3.55259609267310069855347421589, 5.20211818839708393925024617214, 6.29893604586372413452748513616, 7.86887174279636678168802257709, 8.392246336477516777107228911669, 9.283856551390663677174791241507, 10.93572270069765885774357282668, 11.54986432776316599744325871468, 12.22762802199767104581898410127

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