Properties

Label 2-175-175.3-c3-0-48
Degree $2$
Conductor $175$
Sign $-0.633 + 0.774i$
Analytic cond. $10.3253$
Root an. cond. $3.21330$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.66 − 2.56i)2-s + (3.28 − 8.56i)3-s + (−0.554 − 1.24i)4-s + (5.95 + 9.46i)5-s + (−16.4 − 22.7i)6-s + (16.8 − 7.73i)7-s + (20.0 + 3.17i)8-s + (−42.4 − 38.2i)9-s + (34.2 + 0.487i)10-s + (−18.2 − 20.2i)11-s + (−12.4 + 0.654i)12-s + (−30.7 − 15.6i)13-s + (8.18 − 56.0i)14-s + (100. − 19.8i)15-s + (48.8 − 54.2i)16-s + (−33.3 + 41.1i)17-s + ⋯
L(s)  = 1  + (0.589 − 0.907i)2-s + (0.632 − 1.64i)3-s + (−0.0693 − 0.155i)4-s + (0.532 + 0.846i)5-s + (−1.12 − 1.54i)6-s + (0.908 − 0.417i)7-s + (0.886 + 0.140i)8-s + (−1.57 − 1.41i)9-s + (1.08 + 0.0154i)10-s + (−0.500 − 0.556i)11-s + (−0.300 + 0.0157i)12-s + (−0.656 − 0.334i)13-s + (0.156 − 1.07i)14-s + (1.73 − 0.342i)15-s + (0.763 − 0.848i)16-s + (−0.475 + 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.633 + 0.774i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.633 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $-0.633 + 0.774i$
Analytic conductor: \(10.3253\)
Root analytic conductor: \(3.21330\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :3/2),\ -0.633 + 0.774i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.43296 - 3.02304i\)
\(L(\frac12)\) \(\approx\) \(1.43296 - 3.02304i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-5.95 - 9.46i)T \)
7 \( 1 + (-16.8 + 7.73i)T \)
good2 \( 1 + (-1.66 + 2.56i)T + (-3.25 - 7.30i)T^{2} \)
3 \( 1 + (-3.28 + 8.56i)T + (-20.0 - 18.0i)T^{2} \)
11 \( 1 + (18.2 + 20.2i)T + (-139. + 1.32e3i)T^{2} \)
13 \( 1 + (30.7 + 15.6i)T + (1.29e3 + 1.77e3i)T^{2} \)
17 \( 1 + (33.3 - 41.1i)T + (-1.02e3 - 4.80e3i)T^{2} \)
19 \( 1 + (-17.3 - 7.74i)T + (4.58e3 + 5.09e3i)T^{2} \)
23 \( 1 + (-91.8 - 59.6i)T + (4.94e3 + 1.11e4i)T^{2} \)
29 \( 1 + (142. - 195. i)T + (-7.53e3 - 2.31e4i)T^{2} \)
31 \( 1 + (-112. + 11.8i)T + (2.91e4 - 6.19e3i)T^{2} \)
37 \( 1 + (-9.15 - 174. i)T + (-5.03e4 + 5.29e3i)T^{2} \)
41 \( 1 + (190. - 62.0i)T + (5.57e4 - 4.05e4i)T^{2} \)
43 \( 1 + (131. + 131. i)T + 7.95e4iT^{2} \)
47 \( 1 + (-440. + 356. i)T + (2.15e4 - 1.01e5i)T^{2} \)
53 \( 1 + (275. + 105. i)T + (1.10e5 + 9.96e4i)T^{2} \)
59 \( 1 + (-425. - 90.3i)T + (1.87e5 + 8.35e4i)T^{2} \)
61 \( 1 + (-14.1 - 66.3i)T + (-2.07e5 + 9.23e4i)T^{2} \)
67 \( 1 + (-594. - 481. i)T + (6.25e4 + 2.94e5i)T^{2} \)
71 \( 1 + (785. + 570. i)T + (1.10e5 + 3.40e5i)T^{2} \)
73 \( 1 + (540. + 28.3i)T + (3.86e5 + 4.06e4i)T^{2} \)
79 \( 1 + (708. + 74.4i)T + (4.82e5 + 1.02e5i)T^{2} \)
83 \( 1 + (185. - 1.17e3i)T + (-5.43e5 - 1.76e5i)T^{2} \)
89 \( 1 + (-819. + 174. i)T + (6.44e5 - 2.86e5i)T^{2} \)
97 \( 1 + (-223. - 1.40e3i)T + (-8.68e5 + 2.82e5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.99126934667211952927147651536, −11.20935031660512147238079986159, −10.31794993057267973087860459602, −8.553973364580430735564072155702, −7.57271851640837300853678353569, −6.92300486557307239581889292585, −5.35933817903243504753054601367, −3.38382282065750220021560948472, −2.40293800716337947930207709889, −1.41582977540055375978419728647, 2.30251504759891671715868600132, 4.47169788620798254022033094383, 4.80665254007988519792913662249, 5.71945161274789665050283561223, 7.53719216110327275684907093953, 8.643155865632707473246481052059, 9.509874812199559279446020964226, 10.35627630417473940903309990007, 11.48390925648834767798740382903, 13.00411985694400650001629940073

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