Properties

Label 2-175-175.103-c1-0-15
Degree $2$
Conductor $175$
Sign $0.486 + 0.873i$
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  + (2.20 + 0.115i)2-s + (−1.65 − 2.04i)3-s + (2.86 + 0.300i)4-s + (0.294 − 2.21i)5-s + (−3.42 − 4.70i)6-s + (−1.02 + 2.43i)7-s + (1.91 + 0.303i)8-s + (−0.819 + 3.85i)9-s + (0.906 − 4.85i)10-s + (4.32 − 0.918i)11-s + (−4.13 − 6.36i)12-s + (1.80 + 0.921i)13-s + (−2.55 + 5.25i)14-s + (−5.02 + 3.07i)15-s + (−1.43 − 0.305i)16-s + (−0.163 − 0.427i)17-s + ⋯
L(s)  = 1  + (1.55 + 0.0817i)2-s + (−0.957 − 1.18i)3-s + (1.43 + 0.150i)4-s + (0.131 − 0.991i)5-s + (−1.39 − 1.92i)6-s + (−0.388 + 0.921i)7-s + (0.677 + 0.107i)8-s + (−0.273 + 1.28i)9-s + (0.286 − 1.53i)10-s + (1.30 − 0.277i)11-s + (−1.19 − 1.83i)12-s + (0.501 + 0.255i)13-s + (−0.681 + 1.40i)14-s + (−1.29 + 0.793i)15-s + (−0.359 − 0.0764i)16-s + (−0.0397 − 0.103i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.486 + 0.873i$
Analytic conductor: \(1.39738\)
Root analytic conductor: \(1.18210\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :1/2),\ 0.486 + 0.873i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65718 - 0.974191i\)
\(L(\frac12)\) \(\approx\) \(1.65718 - 0.974191i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.294 + 2.21i)T \)
7 \( 1 + (1.02 - 2.43i)T \)
good2 \( 1 + (-2.20 - 0.115i)T + (1.98 + 0.209i)T^{2} \)
3 \( 1 + (1.65 + 2.04i)T + (-0.623 + 2.93i)T^{2} \)
11 \( 1 + (-4.32 + 0.918i)T + (10.0 - 4.47i)T^{2} \)
13 \( 1 + (-1.80 - 0.921i)T + (7.64 + 10.5i)T^{2} \)
17 \( 1 + (0.163 + 0.427i)T + (-12.6 + 11.3i)T^{2} \)
19 \( 1 + (-0.752 - 7.15i)T + (-18.5 + 3.95i)T^{2} \)
23 \( 1 + (0.214 - 4.09i)T + (-22.8 - 2.40i)T^{2} \)
29 \( 1 + (-4.82 + 6.64i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.349 + 0.784i)T + (-20.7 - 23.0i)T^{2} \)
37 \( 1 + (5.20 - 3.37i)T + (15.0 - 33.8i)T^{2} \)
41 \( 1 + (-0.382 + 0.124i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + (1.61 + 1.61i)T + 43iT^{2} \)
47 \( 1 + (8.18 + 3.14i)T + (34.9 + 31.4i)T^{2} \)
53 \( 1 + (-0.148 + 0.119i)T + (11.0 - 51.8i)T^{2} \)
59 \( 1 + (2.29 - 2.55i)T + (-6.16 - 58.6i)T^{2} \)
61 \( 1 + (0.888 - 0.800i)T + (6.37 - 60.6i)T^{2} \)
67 \( 1 + (-4.00 + 1.53i)T + (49.7 - 44.8i)T^{2} \)
71 \( 1 + (-0.544 - 0.395i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-9.23 + 14.2i)T + (-29.6 - 66.6i)T^{2} \)
79 \( 1 + (-5.67 - 12.7i)T + (-52.8 + 58.7i)T^{2} \)
83 \( 1 + (0.785 - 4.96i)T + (-78.9 - 25.6i)T^{2} \)
89 \( 1 + (-0.0463 - 0.0515i)T + (-9.30 + 88.5i)T^{2} \)
97 \( 1 + (-0.650 - 4.10i)T + (-92.2 + 29.9i)T^{2} \)
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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

−12.38668462928429195270899933240, −12.04435190688629480014573553347, −11.50386893967335049115086616252, −9.501072555511154065552285111110, −8.203519623705533733191926190539, −6.57083129440309991402871742510, −6.04025807680433186176604756491, −5.24689942097444462477590327241, −3.77319884725817938126412692706, −1.68051770288351105253500371869, 3.22748514881641655806478085376, 4.11513958604557426680322139313, 5.03357318276450429087243935580, 6.42087827251298531835786965230, 6.80240129204932883019994407647, 9.264488491709079093795842232014, 10.44172863310057013120559959756, 11.03146825029683138647776981782, 11.77115442019766357546797460028, 12.93332777202901160004004975579

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