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Properties

Label 2-175-35.27-c3-0-21
Degree $2$
Conductor $175$
Sign $0.917 + 0.397i$
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  + (2 + 2i)2-s + (−2.23 − 2.23i)3-s − 8.94i·6-s + (2.67 + 18.3i)7-s + (16 − 16i)8-s − 17i·9-s + 24·11-s + (−42.4 − 42.4i)13-s + (−31.3 + 42i)14-s + 64·16-s + (91.6 − 91.6i)17-s + (34 − 34i)18-s + 80.4·19-s + (35 − 46.9i)21-s + (48 + 48i)22-s + (59 − 59i)23-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + (−0.430 − 0.430i)3-s − 0.608i·6-s + (0.144 + 0.989i)7-s + (0.707 − 0.707i)8-s − 0.629i·9-s + 0.657·11-s + (−0.906 − 0.906i)13-s + (−0.597 + 0.801i)14-s + 16-s + (1.30 − 1.30i)17-s + (0.445 − 0.445i)18-s + 0.971·19-s + (0.363 − 0.487i)21-s + (0.465 + 0.465i)22-s + (0.534 − 0.534i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 + 0.397i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.917 + 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.917 + 0.397i$
Analytic conductor: \(10.3253\)
Root analytic conductor: \(3.21330\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (132, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :3/2),\ 0.917 + 0.397i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.20384 - 0.456729i\)
\(L(\frac12)\) \(\approx\) \(2.20384 - 0.456729i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.67 - 18.3i)T \)
good2 \( 1 + (-2 - 2i)T + 8iT^{2} \)
3 \( 1 + (2.23 + 2.23i)T + 27iT^{2} \)
11 \( 1 - 24T + 1.33e3T^{2} \)
13 \( 1 + (42.4 + 42.4i)T + 2.19e3iT^{2} \)
17 \( 1 + (-91.6 + 91.6i)T - 4.91e3iT^{2} \)
19 \( 1 - 80.4T + 6.85e3T^{2} \)
23 \( 1 + (-59 + 59i)T - 1.21e4iT^{2} \)
29 \( 1 - 52iT - 2.43e4T^{2} \)
31 \( 1 - 58.1iT - 2.97e4T^{2} \)
37 \( 1 + (93 + 93i)T + 5.06e4iT^{2} \)
41 \( 1 - 26.8iT - 6.89e4T^{2} \)
43 \( 1 + (221 - 221i)T - 7.95e4iT^{2} \)
47 \( 1 + (51.4 - 51.4i)T - 1.03e5iT^{2} \)
53 \( 1 + (-87 + 87i)T - 1.48e5iT^{2} \)
59 \( 1 - 420.T + 2.05e5T^{2} \)
61 \( 1 - 339. iT - 2.26e5T^{2} \)
67 \( 1 + (-277 - 277i)T + 3.00e5iT^{2} \)
71 \( 1 + 916T + 3.57e5T^{2} \)
73 \( 1 + (78.2 + 78.2i)T + 3.89e5iT^{2} \)
79 \( 1 + 478iT - 4.93e5T^{2} \)
83 \( 1 + (-462. - 462. i)T + 5.71e5iT^{2} \)
89 \( 1 + 4.47T + 7.04e5T^{2} \)
97 \( 1 + (-959. + 959. i)T - 9.12e5iT^{2} \)
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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

−12.23242259383910226408303228106, −11.67431265391238860856675466558, −10.07753720304788639092387939569, −9.204233418610638340522896126512, −7.61743915075636610899737199546, −6.75968893520848310087545576382, −5.63429173100620676129578373683, −5.04164015106336359236003429223, −3.17017921739595577511774286506, −0.980874808875993978604373887776, 1.66361304469990004579170472372, 3.50004788490674053173352356656, 4.42224252059760775132197072062, 5.41998588573014733492031613937, 7.14184562161796785315440842483, 8.074515501334951076682189364634, 9.759687351425455187791383101584, 10.53592891659627835708974156082, 11.50265117644153074573268956800, 12.10560950759206661893627967534

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