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Properties

Label 2-174-87.17-c3-0-27
Degree $2$
Conductor $174$
Sign $-0.498 + 0.866i$
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 + (−3.92 − 3.39i)3-s + 4.00i·4-s − 1.42·5-s + (−0.750 − 10.3i)6-s + 6.13·7-s + (−5.65 + 5.65i)8-s + (3.88 + 26.7i)9-s + (−2.00 − 2.00i)10-s + (−25.3 − 25.3i)11-s + (13.5 − 15.7i)12-s − 63.0i·13-s + (8.67 + 8.67i)14-s + (5.58 + 4.82i)15-s − 16.0·16-s + (−71.2 − 71.2i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.756 − 0.654i)3-s + 0.500i·4-s − 0.127·5-s + (−0.0510 − 0.705i)6-s + 0.331·7-s + (−0.250 + 0.250i)8-s + (0.144 + 0.989i)9-s + (−0.0635 − 0.0635i)10-s + (−0.694 − 0.694i)11-s + (0.327 − 0.378i)12-s − 1.34i·13-s + (0.165 + 0.165i)14-s + (0.0961 + 0.0831i)15-s − 0.250·16-s + (−1.01 − 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 174 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.498 + 0.866i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 174 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.498 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(174\)    =    \(2 \cdot 3 \cdot 29\)
Sign: $-0.498 + 0.866i$
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: \((2,\ 174,\ (\ :3/2),\ -0.498 + 0.866i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.364023 - 0.629152i\)
\(L(\frac12)\) \(\approx\) \(0.364023 - 0.629152i\)
\(L(\frac{5}{2})\) 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 \( 1 + (3.92 + 3.39i)T \)
29 \( 1 + (143. + 61.3i)T \)
good5 \( 1 + 1.42T + 125T^{2} \)
7 \( 1 - 6.13T + 343T^{2} \)
11 \( 1 + (25.3 + 25.3i)T + 1.33e3iT^{2} \)
13 \( 1 + 63.0iT - 2.19e3T^{2} \)
17 \( 1 + (71.2 + 71.2i)T + 4.91e3iT^{2} \)
19 \( 1 + (82.3 - 82.3i)T - 6.85e3iT^{2} \)
23 \( 1 + 134. iT - 1.21e4T^{2} \)
31 \( 1 + (-142. + 142. i)T - 2.97e4iT^{2} \)
37 \( 1 + (-312. - 312. i)T + 5.06e4iT^{2} \)
41 \( 1 + (225. - 225. i)T - 6.89e4iT^{2} \)
43 \( 1 + (-79.9 + 79.9i)T - 7.95e4iT^{2} \)
47 \( 1 + (-361. + 361. i)T - 1.03e5iT^{2} \)
53 \( 1 - 331. iT - 1.48e5T^{2} \)
59 \( 1 - 215. iT - 2.05e5T^{2} \)
61 \( 1 + (-341. + 341. i)T - 2.26e5iT^{2} \)
67 \( 1 - 392. iT - 3.00e5T^{2} \)
71 \( 1 + 59.9T + 3.57e5T^{2} \)
73 \( 1 + (745. + 745. i)T + 3.89e5iT^{2} \)
79 \( 1 + (-76.1 + 76.1i)T - 4.93e5iT^{2} \)
83 \( 1 - 751. iT - 5.71e5T^{2} \)
89 \( 1 + (-19.0 - 19.0i)T + 7.04e5iT^{2} \)
97 \( 1 + (291. + 291. i)T + 9.12e5iT^{2} \)
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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

−12.02288179745768543133150355033, −11.17869144443430163809356828135, −10.21419845674508416566812991945, −8.303086428462847492030127727760, −7.77274565623383540679314965845, −6.41758161522278254907303635620, −5.60788785141832102203652614265, −4.46431386874243930057525448881, −2.54921812486399868548488624622, −0.29215598398706022221878387306, 1.99390738496860964443140637838, 3.99711289643843952423567661670, 4.71477050474432860649177755712, 5.97291025936774945546883030316, 7.11838224328976901382328251949, 8.871061091439612052215523962197, 9.815143227398860385248156400042, 10.95337898115882309471194585101, 11.38378030548031899206929115110, 12.49142187657664326512570719069

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