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Properties

Label 2-170-17.15-c1-0-0
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: $\chi_{170} (151, \cdot )$
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

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

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