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Properties

Label 2-175-35.9-c3-0-27
Degree $2$
Conductor $175$
Sign $-0.998 - 0.0610i$
Analytic cond. $10.3253$
Root an. cond. $3.21330$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.197 − 0.113i)2-s + (−1.56 + 0.904i)3-s + (−3.97 − 6.88i)4-s + 0.411·6-s + (16.6 − 8.20i)7-s + 3.62i·8-s + (−11.8 + 20.5i)9-s + (−8.85 − 15.3i)11-s + (12.4 + 7.18i)12-s − 62.3i·13-s + (−4.20 − 0.271i)14-s + (−31.3 + 54.3i)16-s + (−75.4 + 43.5i)17-s + (4.67 − 2.69i)18-s + (−50.8 + 88.0i)19-s + ⋯
L(s)  = 1  + (−0.0696 − 0.0402i)2-s + (−0.301 + 0.174i)3-s + (−0.496 − 0.860i)4-s + 0.0279·6-s + (0.896 − 0.443i)7-s + 0.160i·8-s + (−0.439 + 0.761i)9-s + (−0.242 − 0.420i)11-s + (0.299 + 0.172i)12-s − 1.32i·13-s + (−0.0802 − 0.00518i)14-s + (−0.490 + 0.849i)16-s + (−1.07 + 0.621i)17-s + (0.0612 − 0.0353i)18-s + (−0.614 + 1.06i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0610i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0610i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $-0.998 - 0.0610i$
Analytic conductor: \(10.3253\)
Root analytic conductor: \(3.21330\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :3/2),\ -0.998 - 0.0610i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.00831033 + 0.271791i\)
\(L(\frac12)\) \(\approx\) \(0.00831033 + 0.271791i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-16.6 + 8.20i)T \)
good2 \( 1 + (0.197 + 0.113i)T + (4 + 6.92i)T^{2} \)
3 \( 1 + (1.56 - 0.904i)T + (13.5 - 23.3i)T^{2} \)
11 \( 1 + (8.85 + 15.3i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 62.3iT - 2.19e3T^{2} \)
17 \( 1 + (75.4 - 43.5i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (50.8 - 88.0i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (81.0 + 46.8i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 297.T + 2.43e4T^{2} \)
31 \( 1 + (45.6 + 79.0i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-243. - 140. i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 271.T + 6.89e4T^{2} \)
43 \( 1 + 7.81iT - 7.95e4T^{2} \)
47 \( 1 + (80.0 + 46.2i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-156. + 90.1i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (49.8 + 86.3i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (217. - 376. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-399. + 230. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 518.T + 3.57e5T^{2} \)
73 \( 1 + (470. - 271. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-119. + 207. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 299. iT - 5.71e5T^{2} \)
89 \( 1 + (-527. + 913. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 288. iT - 9.12e5T^{2} \)
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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

−11.34163163057838496505072318789, −10.72480885895276429498284767172, −10.06449991486838167233863279971, −8.531264938171025100589657001510, −7.893076968487823862382671122412, −6.04117268011339289093362385664, −5.27727765193580942713276935009, −4.12977962733413188062955304589, −1.92409026784255476563088182352, −0.12638435182437419274658111057, 2.22707201202525880712778956549, 4.00692373004389558715164359521, 5.05584871399399048433458150997, 6.60627048329905963964686922626, 7.60849460894682171937023871611, 8.904636113097245755696633767497, 9.286358992909052946453449124354, 11.25223379992738385490454968611, 11.65448070777578219811512262695, 12.67212072192751202940780049356

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