Properties

Label 2-175-175.103-c1-0-6
Degree $2$
Conductor $175$
Sign $0.716 + 0.697i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 − 0.0642i)2-s + (0.158 + 0.195i)3-s + (−0.490 − 0.0515i)4-s + (−2.14 + 0.624i)5-s + (−0.181 − 0.249i)6-s + (2.13 − 1.56i)7-s + (3.02 + 0.478i)8-s + (0.610 − 2.87i)9-s + (2.67 − 0.627i)10-s + (6.21 − 1.32i)11-s + (−0.0675 − 0.104i)12-s + (−3.08 − 1.57i)13-s + (−2.71 + 1.77i)14-s + (−0.462 − 0.321i)15-s + (−2.71 − 0.576i)16-s + (1.38 + 3.61i)17-s + ⋯
L(s)  = 1  + (−0.866 − 0.0454i)2-s + (0.0914 + 0.112i)3-s + (−0.245 − 0.0257i)4-s + (−0.960 + 0.279i)5-s + (−0.0741 − 0.102i)6-s + (0.806 − 0.590i)7-s + (1.06 + 0.169i)8-s + (0.203 − 0.957i)9-s + (0.845 − 0.198i)10-s + (1.87 − 0.398i)11-s + (−0.0195 − 0.0300i)12-s + (−0.856 − 0.436i)13-s + (−0.726 + 0.475i)14-s + (−0.119 − 0.0828i)15-s + (−0.677 − 0.144i)16-s + (0.336 + 0.875i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.716 + 0.697i$
Analytic conductor: \(1.39738\)
Root analytic conductor: \(1.18210\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :1/2),\ 0.716 + 0.697i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.623413 - 0.253186i\)
\(L(\frac12)\) \(\approx\) \(0.623413 - 0.253186i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.14 - 0.624i)T \)
7 \( 1 + (-2.13 + 1.56i)T \)
good2 \( 1 + (1.22 + 0.0642i)T + (1.98 + 0.209i)T^{2} \)
3 \( 1 + (-0.158 - 0.195i)T + (-0.623 + 2.93i)T^{2} \)
11 \( 1 + (-6.21 + 1.32i)T + (10.0 - 4.47i)T^{2} \)
13 \( 1 + (3.08 + 1.57i)T + (7.64 + 10.5i)T^{2} \)
17 \( 1 + (-1.38 - 3.61i)T + (-12.6 + 11.3i)T^{2} \)
19 \( 1 + (0.630 + 6.00i)T + (-18.5 + 3.95i)T^{2} \)
23 \( 1 + (0.0931 - 1.77i)T + (-22.8 - 2.40i)T^{2} \)
29 \( 1 + (-1.19 + 1.63i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.613 + 1.37i)T + (-20.7 - 23.0i)T^{2} \)
37 \( 1 + (6.46 - 4.19i)T + (15.0 - 33.8i)T^{2} \)
41 \( 1 + (-2.44 + 0.795i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + (-4.23 - 4.23i)T + 43iT^{2} \)
47 \( 1 + (2.45 + 0.941i)T + (34.9 + 31.4i)T^{2} \)
53 \( 1 + (-2.37 + 1.91i)T + (11.0 - 51.8i)T^{2} \)
59 \( 1 + (8.64 - 9.59i)T + (-6.16 - 58.6i)T^{2} \)
61 \( 1 + (0.0639 - 0.0575i)T + (6.37 - 60.6i)T^{2} \)
67 \( 1 + (1.23 - 0.475i)T + (49.7 - 44.8i)T^{2} \)
71 \( 1 + (-1.14 - 0.833i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-7.23 + 11.1i)T + (-29.6 - 66.6i)T^{2} \)
79 \( 1 + (-1.79 - 4.03i)T + (-52.8 + 58.7i)T^{2} \)
83 \( 1 + (-0.0358 + 0.226i)T + (-78.9 - 25.6i)T^{2} \)
89 \( 1 + (-8.63 - 9.59i)T + (-9.30 + 88.5i)T^{2} \)
97 \( 1 + (-0.0559 - 0.353i)T + (-92.2 + 29.9i)T^{2} \)
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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

−12.28018625301323317406708461383, −11.46229545911240961169533417458, −10.54775267805287994852209335954, −9.421270731277893155482004210324, −8.623531008189793896455749607390, −7.62446494634888174701025908356, −6.66729095789841649998106046342, −4.59604848344746306246005428209, −3.72126772053863369333731827380, −0.988986092425500048953726510190, 1.63845506802907923426566309168, 4.11782992855211580102522020655, 5.00874953678403521393653338676, 7.07924097901248467716583487478, 7.87485430651206341777300810742, 8.746331846680598457467746178273, 9.553148650646994500044868094296, 10.80305985826987898480929519419, 11.90945042496636084268823857501, 12.44475787895120183445323575712

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