↑

Properties

Label 2-175-35.12-c1-0-4
Degree $2$
Conductor $175$
Sign $0.376 + 0.926i$
Analytic cond. $1.39738$
Root an. cond. $1.18210$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.41 + 0.647i)2-s + (0.583 − 2.17i)3-s + (3.68 − 2.12i)4-s + 5.63i·6-s + (2.61 + 0.373i)7-s + (−3.98 + 3.98i)8-s + (−1.79 − 1.03i)9-s + (−1.62 − 2.81i)11-s + (−2.48 − 9.25i)12-s + (−0.527 − 0.527i)13-s + (−6.56 + 0.792i)14-s + (2.79 − 4.83i)16-s + (1.28 + 0.344i)17-s + (5.01 + 1.34i)18-s + (2.88 − 4.99i)19-s + ⋯
L(s)  = 1  + (−1.70 + 0.457i)2-s + (0.336 − 1.25i)3-s + (1.84 − 1.06i)4-s + 2.30i·6-s + (0.989 + 0.141i)7-s + (−1.40 + 1.40i)8-s + (−0.599 − 0.346i)9-s + (−0.490 − 0.849i)11-s + (−0.716 − 2.67i)12-s + (−0.146 − 0.146i)13-s + (−1.75 + 0.211i)14-s + (0.698 − 1.20i)16-s + (0.311 + 0.0835i)17-s + (1.18 + 0.316i)18-s + (0.661 − 1.14i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.376 + 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.376 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.376 + 0.926i$
Analytic conductor: \(1.39738\)
Root analytic conductor: \(1.18210\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :1/2),\ 0.376 + 0.926i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.525900 - 0.354043i\)
\(L(\frac12)\) \(\approx\) \(0.525900 - 0.354043i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 + (-2.61 - 0.373i)T \)
good2 \( 1 + (2.41 - 0.647i)T + (1.73 - i)T^{2} \)
3 \( 1 + (-0.583 + 2.17i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (1.62 + 2.81i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.527 + 0.527i)T + 13iT^{2} \)
17 \( 1 + (-1.28 - 0.344i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-2.88 + 4.99i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.811 + 3.02i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 0.824iT - 29T^{2} \)
31 \( 1 + (6.49 - 3.75i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.92 - 1.85i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 1.86iT - 41T^{2} \)
43 \( 1 + (-5.75 + 5.75i)T - 43iT^{2} \)
47 \( 1 + (-2.18 - 8.14i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.83 + 0.760i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-5.41 - 9.38i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.26 - 3.61i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.896 - 3.34i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 6.16T + 71T^{2} \)
73 \( 1 + (1.07 - 4.01i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-2.66 - 1.53i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-11.3 - 11.3i)T + 83iT^{2} \)
89 \( 1 + (-1.66 + 2.88i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.53 - 4.53i)T - 97iT^{2} \)
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

−12.34860510144974574525770509604, −11.25503256907009445910942163932, −10.48532836367644616870347809857, −9.034352919335069681426233767007, −8.337278951605411027595506480010, −7.58064173340979258027907353807, −6.84285302200748582101788804828, −5.47911706209283109858047379834, −2.44820515749800657139111743209, −1.06330541337147260161297563845, 1.93005082403785169123093278964, 3.68096835036490663028895064861, 5.18610343940458029817038472782, 7.34038719571440103153713840064, 8.061521348234251208708272986594, 9.165916737097549636910786976550, 9.867891988165863694010780337223, 10.53122069690297376343192148375, 11.39641424710069995594544062275, 12.38647106570348420032346003900

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