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Properties

Label 2-175-25.14-c1-0-2
Degree $2$
Conductor $175$
Sign $0.900 + 0.434i$
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  + (−1.40 − 1.93i)2-s + (2.61 + 0.850i)3-s + (−1.14 + 3.53i)4-s + (−0.290 + 2.21i)5-s + (−2.03 − 6.25i)6-s + i·7-s + (3.91 − 1.27i)8-s + (3.69 + 2.68i)9-s + (4.69 − 2.55i)10-s + (3.41 − 2.48i)11-s + (−6.01 + 8.28i)12-s + (−1.92 + 2.65i)13-s + (1.93 − 1.40i)14-s + (−2.64 + 5.55i)15-s + (−1.93 − 1.40i)16-s + (2.71 − 0.881i)17-s + ⋯
L(s)  = 1  + (−0.994 − 1.36i)2-s + (1.51 + 0.490i)3-s + (−0.574 + 1.76i)4-s + (−0.129 + 0.991i)5-s + (−0.830 − 2.55i)6-s + 0.377i·7-s + (1.38 − 0.449i)8-s + (1.23 + 0.895i)9-s + (1.48 − 0.807i)10-s + (1.02 − 0.748i)11-s + (−1.73 + 2.39i)12-s + (−0.534 + 0.735i)13-s + (0.517 − 0.375i)14-s + (−0.683 + 1.43i)15-s + (−0.484 − 0.352i)16-s + (0.658 − 0.213i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.900 + 0.434i$
Analytic conductor: \(1.39738\)
Root analytic conductor: \(1.18210\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :1/2),\ 0.900 + 0.434i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05529 - 0.241011i\)
\(L(\frac12)\) \(\approx\) \(1.05529 - 0.241011i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.290 - 2.21i)T \)
7 \( 1 - iT \)
good2 \( 1 + (1.40 + 1.93i)T + (-0.618 + 1.90i)T^{2} \)
3 \( 1 + (-2.61 - 0.850i)T + (2.42 + 1.76i)T^{2} \)
11 \( 1 + (-3.41 + 2.48i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (1.92 - 2.65i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.71 + 0.881i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.859 + 2.64i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (5.05 + 6.96i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (-1.22 + 3.77i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.97 - 6.06i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (5.62 - 7.74i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-0.170 - 0.123i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 7.93iT - 43T^{2} \)
47 \( 1 + (3.23 + 1.05i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-2.77 - 0.902i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-4.42 - 3.21i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-10.5 + 7.67i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (1.16 - 0.379i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (-1.12 + 3.45i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-2.42 - 3.33i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.45 + 4.47i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (11.3 - 3.70i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (2.95 - 2.14i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (7.90 + 2.56i)T + (78.4 + 57.0i)T^{2} \)
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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

−12.21773548606947911637779094233, −11.51737435172548102078301240350, −10.32600652129748585977093005423, −9.750114646108111842832298805377, −8.761628912881985535915118424462, −8.221532958069930134938116788940, −6.76150664754358612255002157815, −4.04616173618617474069554552676, −3.09759042769949370447144278238, −2.18653351682298944210253202509, 1.49457248950045898384852417334, 3.89919537106605393061156991649, 5.61127295379563317669375380617, 7.09917926889609569889889568855, 7.82830175210167693661764306732, 8.409431630078573036277849448767, 9.477062039737413037178551275362, 9.870001037139965682496648321710, 12.10994555621471136513228326188, 13.06032067836382219412324838079

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