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Properties

Label 2-175-175.47-c1-0-12
Degree $2$
Conductor $175$
Sign $0.131 - 0.991i$
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  + (−0.135 + 2.58i)2-s + (1.90 − 1.53i)3-s + (−4.67 − 0.491i)4-s + (2.23 + 0.0410i)5-s + (3.72 + 5.12i)6-s + (2.52 − 0.791i)7-s + (1.09 − 6.91i)8-s + (0.619 − 2.91i)9-s + (−0.409 + 5.77i)10-s + (−5.30 + 1.12i)11-s + (−9.64 + 6.26i)12-s + (−0.126 + 0.247i)13-s + (1.70 + 6.63i)14-s + (4.31 − 3.36i)15-s + (8.52 + 1.81i)16-s + (−4.22 + 1.62i)17-s + ⋯
L(s)  = 1  + (−0.0958 + 1.82i)2-s + (1.09 − 0.888i)3-s + (−2.33 − 0.245i)4-s + (0.999 + 0.0183i)5-s + (1.51 + 2.09i)6-s + (0.954 − 0.299i)7-s + (0.387 − 2.44i)8-s + (0.206 − 0.970i)9-s + (−0.129 + 1.82i)10-s + (−1.59 + 0.339i)11-s + (−2.78 + 1.80i)12-s + (−0.0349 + 0.0686i)13-s + (0.455 + 1.77i)14-s + (1.11 − 0.868i)15-s + (2.13 + 0.452i)16-s + (−1.02 + 0.393i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.131 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.131 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.131 - 0.991i$
Analytic conductor: \(1.39738\)
Root analytic conductor: \(1.18210\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :1/2),\ 0.131 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13471 + 0.993983i\)
\(L(\frac12)\) \(\approx\) \(1.13471 + 0.993983i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.23 - 0.0410i)T \)
7 \( 1 + (-2.52 + 0.791i)T \)
good2 \( 1 + (0.135 - 2.58i)T + (-1.98 - 0.209i)T^{2} \)
3 \( 1 + (-1.90 + 1.53i)T + (0.623 - 2.93i)T^{2} \)
11 \( 1 + (5.30 - 1.12i)T + (10.0 - 4.47i)T^{2} \)
13 \( 1 + (0.126 - 0.247i)T + (-7.64 - 10.5i)T^{2} \)
17 \( 1 + (4.22 - 1.62i)T + (12.6 - 11.3i)T^{2} \)
19 \( 1 + (-0.00214 - 0.0204i)T + (-18.5 + 3.95i)T^{2} \)
23 \( 1 + (5.45 + 0.285i)T + (22.8 + 2.40i)T^{2} \)
29 \( 1 + (0.413 - 0.569i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.64 + 3.68i)T + (-20.7 - 23.0i)T^{2} \)
37 \( 1 + (0.0129 + 0.0199i)T + (-15.0 + 33.8i)T^{2} \)
41 \( 1 + (-5.37 + 1.74i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + (4.35 - 4.35i)T - 43iT^{2} \)
47 \( 1 + (-1.23 + 3.21i)T + (-34.9 - 31.4i)T^{2} \)
53 \( 1 + (-2.34 - 2.89i)T + (-11.0 + 51.8i)T^{2} \)
59 \( 1 + (3.75 - 4.17i)T + (-6.16 - 58.6i)T^{2} \)
61 \( 1 + (-1.43 + 1.28i)T + (6.37 - 60.6i)T^{2} \)
67 \( 1 + (5.44 + 14.1i)T + (-49.7 + 44.8i)T^{2} \)
71 \( 1 + (-1.09 - 0.796i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-6.88 - 4.47i)T + (29.6 + 66.6i)T^{2} \)
79 \( 1 + (1.03 + 2.31i)T + (-52.8 + 58.7i)T^{2} \)
83 \( 1 + (-6.74 - 1.06i)T + (78.9 + 25.6i)T^{2} \)
89 \( 1 + (5.48 + 6.09i)T + (-9.30 + 88.5i)T^{2} \)
97 \( 1 + (13.8 - 2.19i)T + (92.2 - 29.9i)T^{2} \)
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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

−13.53169409064087734046400567099, −12.74309291372454170075081754347, −10.56490016558438730514912643220, −9.372892291605804763568477329538, −8.324273474221903357516426020286, −7.84358829496491609723460394086, −6.91822706473938614822966653872, −5.75580982969408797052708499395, −4.62202085965418755679091781155, −2.17828828588322962171898808112, 2.13533267788473035877820132450, 2.86280890548423158072260116128, 4.37567337823825325087702383317, 5.32867216368395338399458652402, 8.150167454198919043330612851602, 8.822235795841998916332308644766, 9.743641555093470447217161478258, 10.43710894650102957753001883248, 11.16063254033729449195563196539, 12.46253456138052517869469766673

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