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Properties

Label 2-175-35.9-c3-0-26
Degree $2$
Conductor $175$
Sign $0.391 + 0.920i$
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  + (1.37 + 0.792i)2-s + (4.54 − 2.62i)3-s + (−2.74 − 4.75i)4-s + 8.31·6-s + (17.8 − 5.10i)7-s − 21.3i·8-s + (0.242 − 0.420i)9-s + (−14.0 − 24.3i)11-s + (−24.9 − 14.3i)12-s + 3.85i·13-s + (28.4 + 7.10i)14-s + (−4.98 + 8.63i)16-s + (33.2 − 19.1i)17-s + (0.666 − 0.384i)18-s + (58.3 − 101. i)19-s + ⋯
L(s)  = 1  + (0.485 + 0.280i)2-s + (0.873 − 0.504i)3-s + (−0.342 − 0.593i)4-s + 0.565·6-s + (0.961 − 0.275i)7-s − 0.945i·8-s + (0.00898 − 0.0155i)9-s + (−0.385 − 0.668i)11-s + (−0.599 − 0.345i)12-s + 0.0823i·13-s + (0.544 + 0.135i)14-s + (−0.0778 + 0.134i)16-s + (0.474 − 0.273i)17-s + (0.00872 − 0.00503i)18-s + (0.704 − 1.22i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.391 + 0.920i$
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: \((2,\ 175,\ (\ :3/2),\ 0.391 + 0.920i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.27927 - 1.50657i\)
\(L(\frac12)\) \(\approx\) \(2.27927 - 1.50657i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-17.8 + 5.10i)T \)
good2 \( 1 + (-1.37 - 0.792i)T + (4 + 6.92i)T^{2} \)
3 \( 1 + (-4.54 + 2.62i)T + (13.5 - 23.3i)T^{2} \)
11 \( 1 + (14.0 + 24.3i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 3.85iT - 2.19e3T^{2} \)
17 \( 1 + (-33.2 + 19.1i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-58.3 + 101. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (152. + 88.2i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 209.T + 2.43e4T^{2} \)
31 \( 1 + (-103. - 179. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-13.5 - 7.83i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 10.5T + 6.89e4T^{2} \)
43 \( 1 - 325. iT - 7.95e4T^{2} \)
47 \( 1 + (-163. - 94.2i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-238. + 137. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-21.9 - 37.9i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (427. - 740. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (472. - 272. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 1.02e3T + 3.57e5T^{2} \)
73 \( 1 + (-218. + 126. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (461. - 799. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 960. iT - 5.71e5T^{2} \)
89 \( 1 + (66.6 - 115. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.02e3iT - 9.12e5T^{2} \)
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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

−12.30761325856879038847415801342, −11.05521058695858773422783186979, −10.05502089131653328297126720467, −8.758053226831317174576573880616, −8.002588325499214794026992393198, −6.84996749295681751455424504315, −5.46634607128395574137785395737, −4.45762632590269416334745843534, −2.77984834948785585312412548276, −1.07645672034891283059352257046, 2.17142415664763979969266511036, 3.49317070162148028743768000800, 4.44657389759900781229885196989, 5.66408049851166994678336979841, 7.81849801858679977144026040333, 8.182572900003667010566216623677, 9.381653907995503172218134379889, 10.34819233633807527062174821556, 11.89524539170641555814294146432, 12.19776214090870061611260648517

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