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Properties

Label 2-175-5.4-c5-0-23
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  + 0.725i·2-s − 19.6i·3-s + 31.4·4-s + 14.2·6-s + 49i·7-s + 46.0i·8-s − 143.·9-s + 666.·11-s − 618. i·12-s + 650. i·13-s − 35.5·14-s + 973.·16-s + 1.18e3i·17-s − 103. i·18-s + 1.56e3·19-s + ⋯
L(s)  = 1  + 0.128i·2-s − 1.26i·3-s + 0.983·4-s + 0.161·6-s + 0.377i·7-s + 0.254i·8-s − 0.588·9-s + 1.65·11-s − 1.23i·12-s + 1.06i·13-s − 0.0484·14-s + 0.950·16-s + 0.996i·17-s − 0.0754i·18-s + 0.994·19-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

\(L(3)\) \(\approx\) \(2.900512951\)
\(L(\frac12)\) \(\approx\) \(2.900512951\)
\(L(\frac{7}{2})\) 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 \( 1 - 0.725iT - 32T^{2} \)
3 \( 1 + 19.6iT - 243T^{2} \)
11 \( 1 - 666.T + 1.61e5T^{2} \)
13 \( 1 - 650. iT - 3.71e5T^{2} \)
17 \( 1 - 1.18e3iT - 1.41e6T^{2} \)
19 \( 1 - 1.56e3T + 2.47e6T^{2} \)
23 \( 1 - 1.10e3iT - 6.43e6T^{2} \)
29 \( 1 + 2.39e3T + 2.05e7T^{2} \)
31 \( 1 + 2.04e3T + 2.86e7T^{2} \)
37 \( 1 - 1.07e3iT - 6.93e7T^{2} \)
41 \( 1 - 1.09e3T + 1.15e8T^{2} \)
43 \( 1 + 1.65e4iT - 1.47e8T^{2} \)
47 \( 1 + 8.29e3iT - 2.29e8T^{2} \)
53 \( 1 + 5.51e3iT - 4.18e8T^{2} \)
59 \( 1 - 1.42e4T + 7.14e8T^{2} \)
61 \( 1 + 1.42e4T + 8.44e8T^{2} \)
67 \( 1 - 1.97e4iT - 1.35e9T^{2} \)
71 \( 1 - 6.45e4T + 1.80e9T^{2} \)
73 \( 1 + 2.85e4iT - 2.07e9T^{2} \)
79 \( 1 - 3.06e4T + 3.07e9T^{2} \)
83 \( 1 - 675. iT - 3.93e9T^{2} \)
89 \( 1 + 1.25e5T + 5.58e9T^{2} \)
97 \( 1 + 2.29e4iT - 8.58e9T^{2} \)
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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

−11.89973685161212664221688674724, −11.18843568944449391200932634389, −9.621901760360368506171618823393, −8.471920739833752295394099375015, −7.25308632207355758057379669385, −6.68838416238483559044529495390, −5.79060854992652268465546655804, −3.73217024238547272792880930843, −2.02188499118541663053174572586, −1.31562820103702237273734432952, 1.11996315233722222587323019420, 3.02417088929904646044511933564, 3.98023680574650254558146979086, 5.32542772183890970907090379396, 6.61659349724714654412557693828, 7.69027436718918716209788719979, 9.256989177431032117317783732331, 9.909956328465065664103438821573, 10.93215217789796996540463027422, 11.56747542842429710770708656783

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