Properties

Label 2-175-35.12-c1-0-3
Degree $2$
Conductor $175$
Sign $0.848 - 0.528i$
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  + (−0.5 + 0.133i)2-s + (0.133 − 0.5i)3-s + (−1.5 + 0.866i)4-s + 0.267i·6-s + (2.5 + 0.866i)7-s + (1.36 − 1.36i)8-s + (2.36 + 1.36i)9-s + (1.36 + 2.36i)11-s + (0.232 + 0.866i)12-s + (2 + 2i)13-s + (−1.36 − 0.0980i)14-s + (1.23 − 2.13i)16-s + (−3.73 − i)17-s + (−1.36 − 0.366i)18-s + (0.366 − 0.633i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.0947i)2-s + (0.0773 − 0.288i)3-s + (−0.750 + 0.433i)4-s + 0.109i·6-s + (0.944 + 0.327i)7-s + (0.482 − 0.482i)8-s + (0.788 + 0.455i)9-s + (0.411 + 0.713i)11-s + (0.0669 + 0.249i)12-s + (0.554 + 0.554i)13-s + (−0.365 − 0.0262i)14-s + (0.308 − 0.533i)16-s + (−0.905 − 0.242i)17-s + (−0.321 − 0.0862i)18-s + (0.0839 − 0.145i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 - 0.528i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.848 - 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.848 - 0.528i$
Analytic conductor: \(1.39738\)
Root analytic conductor: \(1.18210\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :1/2),\ 0.848 - 0.528i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.939704 + 0.268689i\)
\(L(\frac12)\) \(\approx\) \(0.939704 + 0.268689i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 0.866i)T \)
good2 \( 1 + (0.5 - 0.133i)T + (1.73 - i)T^{2} \)
3 \( 1 + (-0.133 + 0.5i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (-1.36 - 2.36i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2 - 2i)T + 13iT^{2} \)
17 \( 1 + (3.73 + i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.366 + 0.633i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.0358 + 0.133i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 3iT - 29T^{2} \)
31 \( 1 + (6.46 - 3.73i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.73 + 1.26i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 6.46iT - 41T^{2} \)
43 \( 1 + (2.83 - 2.83i)T - 43iT^{2} \)
47 \( 1 + (2.36 + 8.83i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (6.83 + 1.83i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (4.09 + 7.09i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.33 - 0.767i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.86 - 10.6i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 1.26T + 71T^{2} \)
73 \( 1 + (-3.46 + 12.9i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-2.83 - 1.63i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.09 - 2.09i)T + 83iT^{2} \)
89 \( 1 + (0.330 - 0.571i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.92 + 5.92i)T - 97iT^{2} \)
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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

−12.88159004503712372725211459890, −11.89691330148397871461389820239, −10.80615907083101180519542025961, −9.555679502925952738676566369936, −8.724633377456072125069331342388, −7.75289709037691972528878450711, −6.83254941459970093343343053223, −4.97703755259523160912737068799, −4.09170186498259751481065436535, −1.81299987570860903987960866303, 1.30726378938097481718546695029, 3.83444985962074491034512330671, 4.83674537912692526457307710184, 6.17686119981450019069870709037, 7.74493522192712570841171594213, 8.712706971656801701001470968287, 9.548081958998004564952538252901, 10.66904077004593931355093531631, 11.25489075690703172988340066736, 12.80129006109182691949330571466

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