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Properties

Label 2-175-7.4-c3-0-28
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: $\chi_{175} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :3/2),\ 0.756 + 0.654i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.14942 - 1.17288i\)
\(L(\frac12)\) \(\approx\) \(3.14942 - 1.17288i\)
\(L(\frac{5}{2})\) 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

−12.41152750402360478317410586035, −11.63379552757649579574950800236, −9.956170516666073696916722924113, −8.455546969243336562804431134788, −7.71775001247906046170871743457, −7.08914698841356124357135691538, −6.02386602858614825435974780659, −4.68805132770038269900287637400, −2.57091068716130840992606843500, −1.38769075209499028136722310905, 2.32414346508022888409811930989, 3.24283406522299603341210494301, 4.57793404528900841322987386915, 4.96301088290189902260332505065, 7.59801356365949939055697873443, 8.462666152686849098944381424862, 9.642907560001706136370363516328, 10.50786891075202608728996377141, 11.15302278484392904623306671721, 12.16841488003012795862859804676

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