Properties

Label 2-170-85.7-c1-0-5
Degree $2$
Conductor $170$
Sign $0.985 + 0.172i$
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 + (0.253 + 1.27i)3-s + (0.707 − 0.707i)4-s + (0.580 − 2.15i)5-s + (0.722 + 1.08i)6-s + (0.192 − 0.128i)7-s + (0.382 − 0.923i)8-s + (1.21 − 0.501i)9-s + (−0.289 − 2.21i)10-s + (−3.12 + 2.08i)11-s + (1.08 + 0.722i)12-s + 6.31i·13-s + (0.128 − 0.192i)14-s + (2.89 + 0.192i)15-s i·16-s + (3.51 − 2.15i)17-s + ⋯
L(s)  = 1  + (0.653 − 0.270i)2-s + (0.146 + 0.735i)3-s + (0.353 − 0.353i)4-s + (0.259 − 0.965i)5-s + (0.294 + 0.441i)6-s + (0.0728 − 0.0486i)7-s + (0.135 − 0.326i)8-s + (0.403 − 0.167i)9-s + (−0.0916 − 0.701i)10-s + (−0.941 + 0.629i)11-s + (0.311 + 0.208i)12-s + 1.75i·13-s + (0.0344 − 0.0514i)14-s + (0.748 + 0.0497i)15-s − 0.250i·16-s + (0.852 − 0.522i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.172i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.985 + 0.172i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.985 + 0.172i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71065 - 0.148526i\)
\(L(\frac12)\) \(\approx\) \(1.71065 - 0.148526i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.580 + 2.15i)T \)
17 \( 1 + (-3.51 + 2.15i)T \)
good3 \( 1 + (-0.253 - 1.27i)T + (-2.77 + 1.14i)T^{2} \)
7 \( 1 + (-0.192 + 0.128i)T + (2.67 - 6.46i)T^{2} \)
11 \( 1 + (3.12 - 2.08i)T + (4.20 - 10.1i)T^{2} \)
13 \( 1 - 6.31iT - 13T^{2} \)
19 \( 1 + (5.28 + 2.18i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (8.43 + 1.67i)T + (21.2 + 8.80i)T^{2} \)
29 \( 1 + (0.221 + 1.11i)T + (-26.7 + 11.0i)T^{2} \)
31 \( 1 + (1.28 + 0.856i)T + (11.8 + 28.6i)T^{2} \)
37 \( 1 + (0.247 - 0.0492i)T + (34.1 - 14.1i)T^{2} \)
41 \( 1 + (-1.06 + 5.36i)T + (-37.8 - 15.6i)T^{2} \)
43 \( 1 + (-8.89 - 3.68i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 4.58T + 47T^{2} \)
53 \( 1 + (-2.39 - 5.77i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-0.547 - 1.32i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (10.1 + 2.02i)T + (56.3 + 23.3i)T^{2} \)
67 \( 1 + (-0.973 + 0.973i)T - 67iT^{2} \)
71 \( 1 + (1.06 - 1.59i)T + (-27.1 - 65.5i)T^{2} \)
73 \( 1 + (-1.60 - 1.07i)T + (27.9 + 67.4i)T^{2} \)
79 \( 1 + (0.466 + 0.698i)T + (-30.2 + 72.9i)T^{2} \)
83 \( 1 + (-13.3 + 5.53i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-9.96 + 9.96i)T - 89iT^{2} \)
97 \( 1 + (-4.18 - 2.79i)T + (37.1 + 89.6i)T^{2} \)
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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

−12.62764510511899939165543171425, −12.04273872499916010790756671147, −10.68853999343415845009980895732, −9.768083261462818911983217396228, −9.009134223084019916388870111007, −7.51521254161019686196283601760, −6.04904405664183390491695842673, −4.66059222779125805278227774585, −4.18002954264777240145140164556, −2.08396029103170665911535333307, 2.31008053851880225259882056984, 3.61033427811202508899576615377, 5.54145533666023185838440123694, 6.29881528454507597046184350376, 7.73185893347043367412004720953, 8.028488197970287879209516508806, 10.25525982618641936150601265393, 10.65932857606861697339661781438, 12.18191204840209327415268711432, 12.94824733565806980536761123258

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