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Properties

Label 2-170-85.12-c1-0-5
Degree $2$
Conductor $170$
Sign $-0.0923 + 0.995i$
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 + (0.442 − 2.22i)3-s + (−0.707 − 0.707i)4-s + (−2.11 − 0.713i)5-s + (1.88 + 1.26i)6-s + (−1.93 − 1.29i)7-s + (0.923 − 0.382i)8-s + (−1.98 − 0.820i)9-s + (1.47 − 1.68i)10-s + (1.86 − 2.78i)11-s + (−1.88 + 1.26i)12-s − 4.73·13-s + (1.93 − 1.29i)14-s + (−2.52 + 4.39i)15-s + i·16-s + (3.36 − 2.38i)17-s + ⋯
L(s)  = 1  + (−0.270 + 0.653i)2-s + (0.255 − 1.28i)3-s + (−0.353 − 0.353i)4-s + (−0.947 − 0.319i)5-s + (0.769 + 0.514i)6-s + (−0.731 − 0.488i)7-s + (0.326 − 0.135i)8-s + (−0.660 − 0.273i)9-s + (0.464 − 0.532i)10-s + (0.561 − 0.840i)11-s + (−0.544 + 0.363i)12-s − 1.31·13-s + (0.517 − 0.345i)14-s + (−0.652 + 1.13i)15-s + 0.250i·16-s + (0.815 − 0.578i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0923 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0923 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $-0.0923 + 0.995i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ -0.0923 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.511616 - 0.561287i\)
\(L(\frac12)\) \(\approx\) \(0.511616 - 0.561287i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.11 + 0.713i)T \)
17 \( 1 + (-3.36 + 2.38i)T \)
good3 \( 1 + (-0.442 + 2.22i)T + (-2.77 - 1.14i)T^{2} \)
7 \( 1 + (1.93 + 1.29i)T + (2.67 + 6.46i)T^{2} \)
11 \( 1 + (-1.86 + 2.78i)T + (-4.20 - 10.1i)T^{2} \)
13 \( 1 + 4.73T + 13T^{2} \)
19 \( 1 + (-0.786 + 0.325i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-4.12 + 0.820i)T + (21.2 - 8.80i)T^{2} \)
29 \( 1 + (-3.83 - 0.762i)T + (26.7 + 11.0i)T^{2} \)
31 \( 1 + (-4.45 - 6.66i)T + (-11.8 + 28.6i)T^{2} \)
37 \( 1 + (9.93 + 1.97i)T + (34.1 + 14.1i)T^{2} \)
41 \( 1 + (0.472 - 0.0940i)T + (37.8 - 15.6i)T^{2} \)
43 \( 1 + (-0.702 - 1.69i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 6.52iT - 47T^{2} \)
53 \( 1 + (-3.22 - 1.33i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (5.52 - 13.3i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (1.17 + 5.90i)T + (-56.3 + 23.3i)T^{2} \)
67 \( 1 + (0.912 - 0.912i)T - 67iT^{2} \)
71 \( 1 + (-10.9 + 7.33i)T + (27.1 - 65.5i)T^{2} \)
73 \( 1 + (-10.1 + 6.77i)T + (27.9 - 67.4i)T^{2} \)
79 \( 1 + (-6.77 - 4.52i)T + (30.2 + 72.9i)T^{2} \)
83 \( 1 + (-3.72 + 8.98i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (2.75 + 2.75i)T + 89iT^{2} \)
97 \( 1 + (7.64 - 5.10i)T + (37.1 - 89.6i)T^{2} \)
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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

−12.44769102504654600791357864480, −11.96293244214826529495541621720, −10.41353295394651746473415186186, −9.103077290575699176791218896105, −8.129695949369995705127337755565, −7.20064680338232907191631569904, −6.69483778090271840624526991698, −5.00567314473245130192232019314, −3.24603288473344647162417733562, −0.78805286170136187030435717132, 2.88233295176820314423318796010, 3.91508587871575195098447519607, 4.95069611439022674010006106500, 6.92234783005754859886606976080, 8.186832609783530539680571383729, 9.463413507064050333439536839012, 9.866770479158790363681557598424, 10.87163367659983855735578440700, 12.08701522806772607787654508738, 12.50920232130417303464918392637

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