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Properties

Label 2-174-87.17-c3-0-2
Degree $2$
Conductor $174$
Sign $-0.948 + 0.315i$
Analytic cond. $10.2663$
Root an. cond. $3.20411$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.41 − 1.41i)2-s + (0.762 + 5.13i)3-s + 4.00i·4-s − 8.24·5-s + (6.19 − 8.34i)6-s + 4.60·7-s + (5.65 − 5.65i)8-s + (−25.8 + 7.83i)9-s + (11.6 + 11.6i)10-s + (12.6 + 12.6i)11-s + (−20.5 + 3.04i)12-s + 35.0i·13-s + (−6.51 − 6.51i)14-s + (−6.28 − 42.3i)15-s − 16.0·16-s + (−90.2 − 90.2i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.146 + 0.989i)3-s + 0.500i·4-s − 0.737·5-s + (0.421 − 0.567i)6-s + 0.248·7-s + (0.250 − 0.250i)8-s + (−0.956 + 0.290i)9-s + (0.368 + 0.368i)10-s + (0.348 + 0.348i)11-s + (−0.494 + 0.0733i)12-s + 0.747i·13-s + (−0.124 − 0.124i)14-s + (−0.108 − 0.729i)15-s − 0.250·16-s + (−1.28 − 1.28i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 174 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 + 0.315i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 174 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.948 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(174\)    =    \(2 \cdot 3 \cdot 29\)
Sign: $-0.948 + 0.315i$
Analytic conductor: \(10.2663\)
Root analytic conductor: \(3.20411\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{174} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 174,\ (\ :3/2),\ -0.948 + 0.315i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.00852316 - 0.0525968i\)
\(L(\frac12)\) \(\approx\) \(0.00852316 - 0.0525968i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 1.41i)T \)
3 \( 1 + (-0.762 - 5.13i)T \)
29 \( 1 + (142. + 63.8i)T \)
good5 \( 1 + 8.24T + 125T^{2} \)
7 \( 1 - 4.60T + 343T^{2} \)
11 \( 1 + (-12.6 - 12.6i)T + 1.33e3iT^{2} \)
13 \( 1 - 35.0iT - 2.19e3T^{2} \)
17 \( 1 + (90.2 + 90.2i)T + 4.91e3iT^{2} \)
19 \( 1 + (17.1 - 17.1i)T - 6.85e3iT^{2} \)
23 \( 1 + 204. iT - 1.21e4T^{2} \)
31 \( 1 + (68.5 - 68.5i)T - 2.97e4iT^{2} \)
37 \( 1 + (-42.8 - 42.8i)T + 5.06e4iT^{2} \)
41 \( 1 + (23.5 - 23.5i)T - 6.89e4iT^{2} \)
43 \( 1 + (-197. + 197. i)T - 7.95e4iT^{2} \)
47 \( 1 + (410. - 410. i)T - 1.03e5iT^{2} \)
53 \( 1 - 242. iT - 1.48e5T^{2} \)
59 \( 1 + 44.1iT - 2.05e5T^{2} \)
61 \( 1 + (-153. + 153. i)T - 2.26e5iT^{2} \)
67 \( 1 + 155. iT - 3.00e5T^{2} \)
71 \( 1 - 325.T + 3.57e5T^{2} \)
73 \( 1 + (-837. - 837. i)T + 3.89e5iT^{2} \)
79 \( 1 + (52.9 - 52.9i)T - 4.93e5iT^{2} \)
83 \( 1 - 358. iT - 5.71e5T^{2} \)
89 \( 1 + (-149. - 149. i)T + 7.04e5iT^{2} \)
97 \( 1 + (735. + 735. i)T + 9.12e5iT^{2} \)
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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

−12.48295845207426151269895738764, −11.42322132103748466888320289760, −11.00010144350532469432741696014, −9.709577740547993153042359199267, −8.982455519359704396605771092937, −8.015255369186305901577859709007, −6.68995975036809987877222750385, −4.74041203521125488534817663711, −3.96900920706141517813652740049, −2.41360960239452379980944609276, 0.02719242588068917748550786074, 1.73254460861449791716171499063, 3.68464959544124802184374883443, 5.54850009054321931020524859766, 6.62974765493256504957463639515, 7.70684597107581773628328079392, 8.308575935414374831360842335209, 9.356976440789880063857676762334, 10.97582148940220783776409849021, 11.55699441037360180304947294124

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