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Properties

Label 2-175-175.108-c1-0-9
Degree $2$
Conductor $175$
Sign $-0.0443 + 0.999i$
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  + (−0.0935 − 1.78i)2-s + (0.279 + 0.226i)3-s + (−1.19 + 0.125i)4-s + (1.07 + 1.95i)5-s + (0.378 − 0.520i)6-s + (−0.314 − 2.62i)7-s + (−0.224 − 1.41i)8-s + (−0.596 − 2.80i)9-s + (3.39 − 2.11i)10-s + (4.12 + 0.877i)11-s + (−0.361 − 0.234i)12-s + (0.268 + 0.527i)13-s + (−4.66 + 0.808i)14-s + (−0.141 + 0.792i)15-s + (−4.85 + 1.03i)16-s + (−0.811 − 0.311i)17-s + ⋯
L(s)  = 1  + (−0.0661 − 1.26i)2-s + (0.161 + 0.130i)3-s + (−0.595 + 0.0626i)4-s + (0.482 + 0.875i)5-s + (0.154 − 0.212i)6-s + (−0.119 − 0.992i)7-s + (−0.0793 − 0.500i)8-s + (−0.198 − 0.935i)9-s + (1.07 − 0.667i)10-s + (1.24 + 0.264i)11-s + (−0.104 − 0.0677i)12-s + (0.0745 + 0.146i)13-s + (−1.24 + 0.215i)14-s + (−0.0365 + 0.204i)15-s + (−1.21 + 0.257i)16-s + (−0.196 − 0.0755i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0443 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0443 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $-0.0443 + 0.999i$
Analytic conductor: \(1.39738\)
Root analytic conductor: \(1.18210\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (108, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :1/2),\ -0.0443 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.893362 - 0.933935i\)
\(L(\frac12)\) \(\approx\) \(0.893362 - 0.933935i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.07 - 1.95i)T \)
7 \( 1 + (0.314 + 2.62i)T \)
good2 \( 1 + (0.0935 + 1.78i)T + (-1.98 + 0.209i)T^{2} \)
3 \( 1 + (-0.279 - 0.226i)T + (0.623 + 2.93i)T^{2} \)
11 \( 1 + (-4.12 - 0.877i)T + (10.0 + 4.47i)T^{2} \)
13 \( 1 + (-0.268 - 0.527i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (0.811 + 0.311i)T + (12.6 + 11.3i)T^{2} \)
19 \( 1 + (0.667 - 6.34i)T + (-18.5 - 3.95i)T^{2} \)
23 \( 1 + (1.61 - 0.0848i)T + (22.8 - 2.40i)T^{2} \)
29 \( 1 + (0.996 + 1.37i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (-3.65 - 8.22i)T + (-20.7 + 23.0i)T^{2} \)
37 \( 1 + (2.37 - 3.65i)T + (-15.0 - 33.8i)T^{2} \)
41 \( 1 + (5.70 + 1.85i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (-3.73 - 3.73i)T + 43iT^{2} \)
47 \( 1 + (3.45 + 8.99i)T + (-34.9 + 31.4i)T^{2} \)
53 \( 1 + (-5.33 + 6.59i)T + (-11.0 - 51.8i)T^{2} \)
59 \( 1 + (-4.53 - 5.03i)T + (-6.16 + 58.6i)T^{2} \)
61 \( 1 + (0.917 + 0.826i)T + (6.37 + 60.6i)T^{2} \)
67 \( 1 + (4.10 - 10.7i)T + (-49.7 - 44.8i)T^{2} \)
71 \( 1 + (11.3 - 8.23i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-2.92 + 1.89i)T + (29.6 - 66.6i)T^{2} \)
79 \( 1 + (-5.74 + 12.9i)T + (-52.8 - 58.7i)T^{2} \)
83 \( 1 + (-3.01 + 0.477i)T + (78.9 - 25.6i)T^{2} \)
89 \( 1 + (-11.4 + 12.6i)T + (-9.30 - 88.5i)T^{2} \)
97 \( 1 + (11.5 + 1.83i)T + (92.2 + 29.9i)T^{2} \)
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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

−12.09945458666828680046465920657, −11.54728698347327874271219772410, −10.23545654822372905978165588560, −10.02972202865390136779398068709, −8.833758903773464827714388731479, −7.00618883127546318871016436728, −6.32663613869579827448238088694, −4.01261488633731249284579447765, −3.30206069371812071122280297726, −1.55934790217064757477056199825, 2.27780110539744026835555132910, 4.69987701092468526192099556245, 5.69368916027785546912160876926, 6.54220747236141164143369944178, 7.903879914395112911169796553397, 8.790865271191678078506469563370, 9.339507226615763152392983851746, 11.12832837418520788470546843633, 12.08343926801369146813089013471, 13.30324502963263853410533236531

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