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Properties

Label 2-175-35.4-c3-0-25
Degree $2$
Conductor $175$
Sign $0.820 + 0.571i$
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  + (−2.42 + 1.39i)2-s + (5.38 + 3.10i)3-s + (−0.0882 + 0.152i)4-s − 17.3·6-s + (−7.92 − 16.7i)7-s − 22.8i·8-s + (5.80 + 10.0i)9-s + (24.8 − 42.9i)11-s + (−0.949 + 0.548i)12-s − 71.6i·13-s + (42.6 + 29.4i)14-s + (31.2 + 54.1i)16-s + (−40.0 − 23.1i)17-s + (−28.1 − 16.2i)18-s + (27.0 + 46.9i)19-s + ⋯
L(s)  = 1  + (−0.856 + 0.494i)2-s + (1.03 + 0.597i)3-s + (−0.0110 + 0.0191i)4-s − 1.18·6-s + (−0.427 − 0.903i)7-s − 1.01i·8-s + (0.215 + 0.372i)9-s + (0.679 − 1.17i)11-s + (−0.0228 + 0.0131i)12-s − 1.52i·13-s + (0.813 + 0.562i)14-s + (0.488 + 0.846i)16-s + (−0.572 − 0.330i)17-s + (−0.368 − 0.212i)18-s + (0.327 + 0.566i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 + 0.571i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.820 + 0.571i$
Analytic conductor: \(10.3253\)
Root analytic conductor: \(3.21330\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :3/2),\ 0.820 + 0.571i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.04137 - 0.326917i\)
\(L(\frac12)\) \(\approx\) \(1.04137 - 0.326917i\)
\(L(\frac{5}{2})\) 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 \( 1 + (7.92 + 16.7i)T \)
good2 \( 1 + (2.42 - 1.39i)T + (4 - 6.92i)T^{2} \)
3 \( 1 + (-5.38 - 3.10i)T + (13.5 + 23.3i)T^{2} \)
11 \( 1 + (-24.8 + 42.9i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 71.6iT - 2.19e3T^{2} \)
17 \( 1 + (40.0 + 23.1i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-27.0 - 46.9i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (132. - 76.4i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 38.5T + 2.43e4T^{2} \)
31 \( 1 + (51.7 - 89.5i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-46.3 + 26.7i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 40.4T + 6.89e4T^{2} \)
43 \( 1 + 377. iT - 7.95e4T^{2} \)
47 \( 1 + (-337. + 194. i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-284. - 164. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (19.8 - 34.3i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (98.7 + 171. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (841. + 486. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 386.T + 3.57e5T^{2} \)
73 \( 1 + (190. + 109. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (193. + 334. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 1.37e3iT - 5.71e5T^{2} \)
89 \( 1 + (-375. - 649. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 533. iT - 9.12e5T^{2} \)
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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

−12.22400644923541565168431050869, −10.63397454676237434229369578269, −9.885343817495709011747747517997, −8.986817726548034749538304799036, −8.223481710278993677704422631561, −7.36622436226642801802918060813, −6.00100690718437909202993406462, −3.91094129665850077591893414453, −3.27660466179752971474590250894, −0.58732824258271721515219233850, 1.76099363407170064375023890682, 2.47437692783953506759059363422, 4.45343431312755568111561941574, 6.25815875801093544343095028571, 7.45131986349357209273433158127, 8.703786158737335618264603500074, 9.147116983431769715977763959801, 9.961801441303697699689679360567, 11.42296129316934903272609446112, 12.19561248207340615533418171016

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