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Properties

Label 2-175-5.3-c4-0-31
Degree $2$
Conductor $175$
Sign $-0.973 - 0.229i$
Analytic cond. $18.0897$
Root an. cond. $4.25320$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.63 − 3.63i)2-s + (−2.99 − 2.99i)3-s − 10.4i·4-s − 21.7·6-s + (13.0 − 13.0i)7-s + (20.2 + 20.2i)8-s − 63.0i·9-s − 220.·11-s + (−31.2 + 31.2i)12-s + (−176. − 176. i)13-s − 95.2i·14-s + 314.·16-s + (101. − 101. i)17-s + (−229. − 229. i)18-s + 152. i·19-s + ⋯
L(s)  = 1  + (0.908 − 0.908i)2-s + (−0.332 − 0.332i)3-s − 0.652i·4-s − 0.604·6-s + (0.267 − 0.267i)7-s + (0.316 + 0.316i)8-s − 0.778i·9-s − 1.81·11-s + (−0.216 + 0.216i)12-s + (−1.04 − 1.04i)13-s − 0.485i·14-s + 1.22·16-s + (0.350 − 0.350i)17-s + (−0.707 − 0.707i)18-s + 0.423i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $-0.973 - 0.229i$
Analytic conductor: \(18.0897\)
Root analytic conductor: \(4.25320\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :2),\ -0.973 - 0.229i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.546513093\)
\(L(\frac12)\) \(\approx\) \(1.546513093\)
\(L(3)\) 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 \( 1 + (-13.0 + 13.0i)T \)
good2 \( 1 + (-3.63 + 3.63i)T - 16iT^{2} \)
3 \( 1 + (2.99 + 2.99i)T + 81iT^{2} \)
11 \( 1 + 220.T + 1.46e4T^{2} \)
13 \( 1 + (176. + 176. i)T + 2.85e4iT^{2} \)
17 \( 1 + (-101. + 101. i)T - 8.35e4iT^{2} \)
19 \( 1 - 152. iT - 1.30e5T^{2} \)
23 \( 1 + (596. + 596. i)T + 2.79e5iT^{2} \)
29 \( 1 - 801. iT - 7.07e5T^{2} \)
31 \( 1 + 175.T + 9.23e5T^{2} \)
37 \( 1 + (423. - 423. i)T - 1.87e6iT^{2} \)
41 \( 1 - 919.T + 2.82e6T^{2} \)
43 \( 1 + (628. + 628. i)T + 3.41e6iT^{2} \)
47 \( 1 + (-2.20e3 + 2.20e3i)T - 4.87e6iT^{2} \)
53 \( 1 + (554. + 554. i)T + 7.89e6iT^{2} \)
59 \( 1 + 3.16e3iT - 1.21e7T^{2} \)
61 \( 1 - 5.29e3T + 1.38e7T^{2} \)
67 \( 1 + (-1.09e3 + 1.09e3i)T - 2.01e7iT^{2} \)
71 \( 1 + 1.62e3T + 2.54e7T^{2} \)
73 \( 1 + (1.90e3 + 1.90e3i)T + 2.83e7iT^{2} \)
79 \( 1 + 5.03e3iT - 3.89e7T^{2} \)
83 \( 1 + (-2.98e3 - 2.98e3i)T + 4.74e7iT^{2} \)
89 \( 1 + 9.63e3iT - 6.27e7T^{2} \)
97 \( 1 + (-2.61e3 + 2.61e3i)T - 8.85e7iT^{2} \)
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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

−11.82048578479441614260336562616, −10.59226564485220922801833245215, −10.10600212428037046174859402724, −8.214892178443291345156565998468, −7.34645884954084445174250013864, −5.65963898008332398740509108809, −4.86811905048077017913932408874, −3.38439906335514537922970917299, −2.27055441157272938935611998111, −0.41249308854171648327305921715, 2.30493606718066228649160008548, 4.26503513673868056034155972398, 5.15846748067586979495029288831, 5.83298528180302925651138843346, 7.36476739491178122480564598339, 7.966546538924397852163891527114, 9.733076285498815493181853543114, 10.56101345231871377898117107165, 11.71791294828997208643132024805, 12.86015095138646728385956330043

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