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Properties

Label 2-175-7.2-c3-0-4
Degree $2$
Conductor $175$
Sign $0.756 - 0.654i$
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  + (−1.46 + 2.52i)2-s + (−5.02 − 8.70i)3-s + (−0.265 − 0.460i)4-s + 29.3·6-s + (−18.3 + 2.32i)7-s − 21.8·8-s + (−36.9 + 64.0i)9-s + (−7.79 − 13.5i)11-s + (−2.67 + 4.62i)12-s + 63.9·13-s + (20.9 − 49.8i)14-s + (33.9 − 58.8i)16-s + (5.28 + 9.16i)17-s + (−108. − 187. i)18-s + (19.3 − 33.5i)19-s + ⋯
L(s)  = 1  + (−0.516 + 0.894i)2-s + (−0.966 − 1.67i)3-s + (−0.0332 − 0.0575i)4-s + 1.99·6-s + (−0.992 + 0.125i)7-s − 0.964·8-s + (−1.36 + 2.37i)9-s + (−0.213 − 0.370i)11-s + (−0.0642 + 0.111i)12-s + 1.36·13-s + (0.399 − 0.952i)14-s + (0.531 − 0.919i)16-s + (0.0754 + 0.130i)17-s + (−1.41 − 2.45i)18-s + (0.234 − 0.405i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 - 0.654i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.756 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.756 - 0.654i$
Analytic conductor: \(10.3253\)
Root analytic conductor: \(3.21330\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :3/2),\ 0.756 - 0.654i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.616867 + 0.229728i\)
\(L(\frac12)\) \(\approx\) \(0.616867 + 0.229728i\)
\(L(\frac{5}{2})\) 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 \( 1 + (18.3 - 2.32i)T \)
good2 \( 1 + (1.46 - 2.52i)T + (-4 - 6.92i)T^{2} \)
3 \( 1 + (5.02 + 8.70i)T + (-13.5 + 23.3i)T^{2} \)
11 \( 1 + (7.79 + 13.5i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 63.9T + 2.19e3T^{2} \)
17 \( 1 + (-5.28 - 9.16i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-19.3 + 33.5i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (12.0 - 20.8i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 226.T + 2.43e4T^{2} \)
31 \( 1 + (-71.5 - 123. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (142. - 247. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 32.0T + 6.89e4T^{2} \)
43 \( 1 - 235.T + 7.95e4T^{2} \)
47 \( 1 + (-72.1 + 124. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-159. - 275. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (224. + 388. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (83.3 - 144. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-370. - 641. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 373.T + 3.57e5T^{2} \)
73 \( 1 + (-118. - 205. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-232. + 403. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 691.T + 5.71e5T^{2} \)
89 \( 1 + (506. - 877. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.15e3T + 9.12e5T^{2} \)
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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

−12.35993774296063654333470135383, −11.66001359586335822240982587103, −10.51643528511410412927706486914, −8.790735817794822630703936023581, −8.037998163241625497976981830950, −6.90250293562052193784605414334, −6.38210297346970064486576081345, −5.58316473018203438290350028982, −2.91152305149920883493789263335, −0.879726789060214576256137004007, 0.57645789922844097698169088710, 3.10949125164753470276744058459, 4.12960736073333316493968589858, 5.67886546933663614752100140684, 6.40498879934222520538925379055, 8.761591554006823229888899912321, 9.567530557959266248869012829239, 10.29370040655133190316208906973, 10.83080758116884963710386901060, 11.75657078644585744747299373732

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