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Properties

Label 2-175-5.4-c5-0-4
Degree $2$
Conductor $175$
Sign $0.894 + 0.447i$
Analytic cond. $28.0671$
Root an. cond. $5.29784$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8.27i·2-s + 25.6i·3-s − 36.4·4-s − 212.·6-s + 49i·7-s − 37.0i·8-s − 414.·9-s − 270.·11-s − 935. i·12-s − 300. i·13-s − 405.·14-s − 860.·16-s + 613. i·17-s − 3.43e3i·18-s + 1.70e3·19-s + ⋯
L(s)  = 1  + 1.46i·2-s + 1.64i·3-s − 1.13·4-s − 2.40·6-s + 0.377i·7-s − 0.204i·8-s − 1.70·9-s − 0.673·11-s − 1.87i·12-s − 0.493i·13-s − 0.552·14-s − 0.840·16-s + 0.514i·17-s − 2.49i·18-s + 1.08·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(28.0671\)
Root analytic conductor: \(5.29784\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :5/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.8071605880\)
\(L(\frac12)\) \(\approx\) \(0.8071605880\)
\(L(\frac{7}{2})\) 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 \( 1 - 49iT \)
good2 \( 1 - 8.27iT - 32T^{2} \)
3 \( 1 - 25.6iT - 243T^{2} \)
11 \( 1 + 270.T + 1.61e5T^{2} \)
13 \( 1 + 300. iT - 3.71e5T^{2} \)
17 \( 1 - 613. iT - 1.41e6T^{2} \)
19 \( 1 - 1.70e3T + 2.47e6T^{2} \)
23 \( 1 + 3.18e3iT - 6.43e6T^{2} \)
29 \( 1 + 4.29e3T + 2.05e7T^{2} \)
31 \( 1 - 2.02e3T + 2.86e7T^{2} \)
37 \( 1 - 5.15e3iT - 6.93e7T^{2} \)
41 \( 1 + 7.14e3T + 1.15e8T^{2} \)
43 \( 1 - 1.95e4iT - 1.47e8T^{2} \)
47 \( 1 - 1.99e4iT - 2.29e8T^{2} \)
53 \( 1 + 3.94e3iT - 4.18e8T^{2} \)
59 \( 1 - 2.97e4T + 7.14e8T^{2} \)
61 \( 1 + 5.05e4T + 8.44e8T^{2} \)
67 \( 1 - 5.05e3iT - 1.35e9T^{2} \)
71 \( 1 - 3.28e4T + 1.80e9T^{2} \)
73 \( 1 - 1.11e4iT - 2.07e9T^{2} \)
79 \( 1 + 8.18e4T + 3.07e9T^{2} \)
83 \( 1 + 1.18e5iT - 3.93e9T^{2} \)
89 \( 1 - 4.16e4T + 5.58e9T^{2} \)
97 \( 1 - 4.36e4iT - 8.58e9T^{2} \)
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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

−13.01850915445719034752691842037, −11.48855466034233747024999449531, −10.48210781141559975756454126206, −9.548767521558910495323003118401, −8.600291330203699525720279327309, −7.73358857736965973562786077151, −6.18774542718380142226486971078, −5.29522454686024614241209559123, −4.50489662729762552978702557329, −2.98495650882085153740970856196, 0.26095649717168051161530436504, 1.37031054122088129466497359234, 2.33162275356981750289669815764, 3.55482997455285191228931430558, 5.38155509493841973191704957488, 6.93467580918626380236630044543, 7.63834038668581608499324708238, 9.017335446730545401254791259322, 10.11918089434484023205093207498, 11.33231484497451138146351427740

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