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Properties

Label 2-175-7.4-c1-0-2
Degree $2$
Conductor $175$
Sign $-0.749 - 0.661i$
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  + (1.20 + 2.09i)2-s + (−0.207 + 0.358i)3-s + (−1.91 + 3.31i)4-s − 6-s + (−2.62 − 0.358i)7-s − 4.41·8-s + (1.41 + 2.44i)9-s + (0.414 − 0.717i)11-s + (−0.792 − 1.37i)12-s + 4.82·13-s + (−2.41 − 5.91i)14-s + (−1.49 − 2.59i)16-s + (2.41 − 4.18i)17-s + (−3.41 + 5.91i)18-s + (−1.41 − 2.44i)19-s + ⋯
L(s)  = 1  + (0.853 + 1.47i)2-s + (−0.119 + 0.207i)3-s + (−0.957 + 1.65i)4-s − 0.408·6-s + (−0.990 − 0.135i)7-s − 1.56·8-s + (0.471 + 0.816i)9-s + (0.124 − 0.216i)11-s + (−0.228 − 0.396i)12-s + 1.33·13-s + (−0.645 − 1.58i)14-s + (−0.374 − 0.649i)16-s + (0.585 − 1.01i)17-s + (−0.804 + 1.39i)18-s + (−0.324 − 0.561i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $-0.749 - 0.661i$
Analytic conductor: \(1.39738\)
Root analytic conductor: \(1.18210\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :1/2),\ -0.749 - 0.661i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.552645 + 1.46162i\)
\(L(\frac12)\) \(\approx\) \(0.552645 + 1.46162i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.62 + 0.358i)T \)
good2 \( 1 + (-1.20 - 2.09i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.207 - 0.358i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-0.414 + 0.717i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.82T + 13T^{2} \)
17 \( 1 + (-2.41 + 4.18i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.41 + 2.44i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.207 - 0.358i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 + (-3 + 5.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.82T + 41T^{2} \)
43 \( 1 + 3.58T + 43T^{2} \)
47 \( 1 + (-1 - 1.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.585 - 1.01i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.24 - 3.88i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.74 + 4.75i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.79 + 8.30i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.48T + 71T^{2} \)
73 \( 1 + (0.414 - 0.717i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.41 + 12.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 13.7T + 83T^{2} \)
89 \( 1 + (-4.32 - 7.49i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 11.6T + 97T^{2} \)
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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

−13.55637308447354758817200424996, −12.61705408573711117372192135701, −11.23384408395969491855003945052, −9.962753378421871246988947346957, −8.724372711696904480656613429132, −7.61376866886425519954578900839, −6.65628991099025667557913204667, −5.75327466049295665616040899848, −4.59585027681082808139750843716, −3.40547581335048268628475731018, 1.43696065403407671941128734596, 3.29905384986511619460563763128, 4.02062679772013825390819254827, 5.72203833901601097691517284851, 6.65994446408115717263248017646, 8.609283637749227736513072157875, 9.831130823246862442859437244010, 10.44247789043968562297362305545, 11.60459311589550346954665165350, 12.47080872374781124850280279307

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