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Properties

Label 2-175-35.9-c3-0-3
Degree $2$
Conductor $175$
Sign $0.669 - 0.742i$
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  + (−2.59 − 1.5i)2-s + (1.73 − i)3-s + (0.5 + 0.866i)4-s − 6·6-s + (−12.1 − 14i)7-s + 21i·8-s + (−11.5 + 19.9i)9-s + (22.5 + 38.9i)11-s + (1.73 + 1.00i)12-s − 59i·13-s + (10.5 + 54.5i)14-s + (35.5 − 61.4i)16-s + (−46.7 + 27i)17-s + (59.7 − 34.5i)18-s + (−60.5 + 104. i)19-s + ⋯
L(s)  = 1  + (−0.918 − 0.530i)2-s + (0.333 − 0.192i)3-s + (0.0625 + 0.108i)4-s − 0.408·6-s + (−0.654 − 0.755i)7-s + 0.928i·8-s + (−0.425 + 0.737i)9-s + (0.616 + 1.06i)11-s + (0.0416 + 0.0240i)12-s − 1.25i·13-s + (0.200 + 1.04i)14-s + (0.554 − 0.960i)16-s + (−0.667 + 0.385i)17-s + (0.782 − 0.451i)18-s + (−0.730 + 1.26i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 - 0.742i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.669 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.669 - 0.742i$
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: \((2,\ 175,\ (\ :3/2),\ 0.669 - 0.742i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.563868 + 0.250811i\)
\(L(\frac12)\) \(\approx\) \(0.563868 + 0.250811i\)
\(L(\frac{5}{2})\) 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 \( 1 + (12.1 + 14i)T \)
good2 \( 1 + (2.59 + 1.5i)T + (4 + 6.92i)T^{2} \)
3 \( 1 + (-1.73 + i)T + (13.5 - 23.3i)T^{2} \)
11 \( 1 + (-22.5 - 38.9i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 59iT - 2.19e3T^{2} \)
17 \( 1 + (46.7 - 27i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (60.5 - 104. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-59.7 - 34.5i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 162T + 2.43e4T^{2} \)
31 \( 1 + (-44 - 76.2i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-224. - 129.5i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 195T + 6.89e4T^{2} \)
43 \( 1 - 286iT - 7.95e4T^{2} \)
47 \( 1 + (38.9 + 22.5i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (517. - 298.5i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (180 + 311. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (196 - 339. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (242. - 140i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 48T + 3.57e5T^{2} \)
73 \( 1 + (578. - 334i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-391 + 677. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 768iT - 5.71e5T^{2} \)
89 \( 1 + (597 - 1.03e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 902iT - 9.12e5T^{2} \)
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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

−12.36775377978531296943805471383, −10.97628738449161312161544806072, −10.33057087875985376169367474969, −9.536191286547175278650072720801, −8.372026367714232313728697917052, −7.59963525531400141588979134749, −6.15871435672187466745397182189, −4.58430884525471169418616116709, −2.84725300216554685714252044493, −1.37584895120259891809060741339, 0.39046274048992051658667993163, 2.85057492030130992935031571010, 4.21267453717774901947659681785, 6.26672639852028589878260860366, 6.76872735471695772447324712228, 8.435769808133281364593774830323, 9.089807402808246174352344020339, 9.429621094690656557612325215932, 11.07168381316520743579832822394, 12.03061189006822718553658135072

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