Properties

Label 2-170-85.37-c1-0-8
Degree $2$
Conductor $170$
Sign $0.238 + 0.971i$
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 + (2.33 − 1.56i)3-s + (0.707 + 0.707i)4-s + (0.438 − 2.19i)5-s + (−2.75 + 0.548i)6-s + (−1.27 + 0.254i)7-s + (−0.382 − 0.923i)8-s + (1.88 − 4.54i)9-s + (−1.24 + 1.85i)10-s + (1.14 + 5.73i)11-s + (2.75 + 0.548i)12-s − 5.33·13-s + (1.27 + 0.254i)14-s + (−2.40 − 5.81i)15-s + i·16-s + (3.93 + 1.22i)17-s + ⋯
L(s)  = 1  + (−0.653 − 0.270i)2-s + (1.35 − 0.902i)3-s + (0.353 + 0.353i)4-s + (0.195 − 0.980i)5-s + (−1.12 + 0.224i)6-s + (−0.483 + 0.0961i)7-s + (−0.135 − 0.326i)8-s + (0.626 − 1.51i)9-s + (−0.393 + 0.587i)10-s + (0.343 + 1.72i)11-s + (0.796 + 0.158i)12-s − 1.48·13-s + (0.341 + 0.0679i)14-s + (−0.620 − 1.50i)15-s + 0.250i·16-s + (0.954 + 0.296i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.238 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.238 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.238 + 0.971i$
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: \((2,\ 170,\ (\ :1/2),\ 0.238 + 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.980132 - 0.768329i\)
\(L(\frac12)\) \(\approx\) \(0.980132 - 0.768329i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.438 + 2.19i)T \)
17 \( 1 + (-3.93 - 1.22i)T \)
good3 \( 1 + (-2.33 + 1.56i)T + (1.14 - 2.77i)T^{2} \)
7 \( 1 + (1.27 - 0.254i)T + (6.46 - 2.67i)T^{2} \)
11 \( 1 + (-1.14 - 5.73i)T + (-10.1 + 4.20i)T^{2} \)
13 \( 1 + 5.33T + 13T^{2} \)
19 \( 1 + (-0.311 - 0.752i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (-3.88 + 5.82i)T + (-8.80 - 21.2i)T^{2} \)
29 \( 1 + (-1.17 - 1.75i)T + (-11.0 + 26.7i)T^{2} \)
31 \( 1 + (-0.301 + 1.51i)T + (-28.6 - 11.8i)T^{2} \)
37 \( 1 + (-3.87 - 5.79i)T + (-14.1 + 34.1i)T^{2} \)
41 \( 1 + (4.40 - 6.59i)T + (-15.6 - 37.8i)T^{2} \)
43 \( 1 + (1.31 - 0.544i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + 0.109iT - 47T^{2} \)
53 \( 1 + (-0.379 + 0.915i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-5.08 - 2.10i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (4.82 + 3.22i)T + (23.3 + 56.3i)T^{2} \)
67 \( 1 + (-4.12 + 4.12i)T - 67iT^{2} \)
71 \( 1 + (5.38 + 1.07i)T + (65.5 + 27.1i)T^{2} \)
73 \( 1 + (8.02 + 1.59i)T + (67.4 + 27.9i)T^{2} \)
79 \( 1 + (-5.01 + 0.997i)T + (72.9 - 30.2i)T^{2} \)
83 \( 1 + (2.38 + 0.987i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (0.779 + 0.779i)T + 89iT^{2} \)
97 \( 1 + (14.7 + 2.93i)T + (89.6 + 37.1i)T^{2} \)
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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

−12.47637161704044358425610427341, −12.15863926819659554280409935007, −9.880325251405263162707898190716, −9.579494741321934067213352937565, −8.467909440304280706388057855412, −7.62044266235825346453254056799, −6.73697523592452360433396138701, −4.64432210079724794781083725691, −2.81286156546384257107084285085, −1.61172105731065870215236710378, 2.73890190770566159548839626620, 3.53190302477128594446685671853, 5.53260190291148986986822277759, 7.03685301409948673921810070124, 7.970977505190393502850386542700, 9.129717765317161699049370772541, 9.751637210898399293855055911978, 10.55777779122099800626671584569, 11.65184968855259588495051960009, 13.47141730015086612550462394147

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