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Properties

Label 2-170-17.8-c1-0-3
Degree $2$
Conductor $170$
Sign $0.583 + 0.811i$
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  + (−0.707 + 0.707i)2-s + (1.10 − 2.66i)3-s − 1.00i·4-s + (0.923 + 0.382i)5-s + (1.10 + 2.66i)6-s + (−0.0470 + 0.0194i)7-s + (0.707 + 0.707i)8-s + (−3.75 − 3.75i)9-s + (−0.923 + 0.382i)10-s + (0.307 + 0.743i)11-s + (−2.66 − 1.10i)12-s − 4.26i·13-s + (0.0194 − 0.0470i)14-s + (2.03 − 2.03i)15-s − 1.00·16-s + (2.43 − 3.32i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.636 − 1.53i)3-s − 0.500i·4-s + (0.413 + 0.171i)5-s + (0.450 + 1.08i)6-s + (−0.0177 + 0.00736i)7-s + (0.250 + 0.250i)8-s + (−1.25 − 1.25i)9-s + (−0.292 + 0.121i)10-s + (0.0928 + 0.224i)11-s + (−0.768 − 0.318i)12-s − 1.18i·13-s + (0.00521 − 0.0125i)14-s + (0.526 − 0.526i)15-s − 0.250·16-s + (0.589 − 0.807i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.583 + 0.811i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.583 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $0.583 + 0.811i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ 0.583 + 0.811i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00378 - 0.514592i\)
\(L(\frac12)\) \(\approx\) \(1.00378 - 0.514592i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (-0.923 - 0.382i)T \)
17 \( 1 + (-2.43 + 3.32i)T \)
good3 \( 1 + (-1.10 + 2.66i)T + (-2.12 - 2.12i)T^{2} \)
7 \( 1 + (0.0470 - 0.0194i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (-0.307 - 0.743i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + 4.26iT - 13T^{2} \)
19 \( 1 + (4.53 - 4.53i)T - 19iT^{2} \)
23 \( 1 + (-3.28 - 7.94i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (-4.22 - 1.74i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (0.189 - 0.456i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (-0.598 + 1.44i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (7.95 - 3.29i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (-1.73 - 1.73i)T + 43iT^{2} \)
47 \( 1 - 8.22iT - 47T^{2} \)
53 \( 1 + (-7.79 + 7.79i)T - 53iT^{2} \)
59 \( 1 + (-7.13 - 7.13i)T + 59iT^{2} \)
61 \( 1 + (5.73 - 2.37i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + 0.822T + 67T^{2} \)
71 \( 1 + (-4.17 + 10.0i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (6.19 + 2.56i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (3.14 + 7.59i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (0.955 - 0.955i)T - 83iT^{2} \)
89 \( 1 - 17.0iT - 89T^{2} \)
97 \( 1 + (-2.95 - 1.22i)T + (68.5 + 68.5i)T^{2} \)
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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

−12.84605075232476612125572682832, −11.86456058815288171218458581910, −10.43796584817495533204618237238, −9.332936304217948410965715036238, −8.196699789367354289431387810666, −7.53684221611807908743618417985, −6.56775087292169688244473827815, −5.50708719406736682725578301598, −2.97996733739649117076777464069, −1.41454334829303442353464486078, 2.45527571290702436286840066973, 3.89767838821113505100559978046, 4.85370249421076868733338516836, 6.63229003810346678034683534522, 8.601939505941634414010430365802, 8.814017999416134368525780270385, 10.01781133679447388301152929318, 10.54780970509608502960671945735, 11.60105422447455780538154578291, 12.91364911305399727429821984824

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