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Properties

Label 2-275-5.4-c3-0-35
Degree $2$
Conductor $275$
Sign $-0.447 + 0.894i$
Analytic cond. $16.2255$
Root an. cond. $4.02809$
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  − 1.43i·2-s − 0.561i·3-s + 5.93·4-s − 0.807·6-s − 31.0i·7-s − 20.0i·8-s + 26.6·9-s − 11·11-s − 3.33i·12-s + 45.6i·13-s − 44.6·14-s + 18.6·16-s − 40.4i·17-s − 38.3i·18-s − 91.2·19-s + ⋯
L(s)  = 1  − 0.508i·2-s − 0.108i·3-s + 0.741·4-s − 0.0549·6-s − 1.67i·7-s − 0.885i·8-s + 0.988·9-s − 0.301·11-s − 0.0801i·12-s + 0.973i·13-s − 0.852·14-s + 0.290·16-s − 0.577i·17-s − 0.502i·18-s − 1.10·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(275\)    =    \(5^{2} \cdot 11\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(16.2255\)
Root analytic conductor: \(4.02809\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{275} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 275,\ (\ :3/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.180296465\)
\(L(\frac12)\) \(\approx\) \(2.180296465\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 + 11T \)
good2 \( 1 + 1.43iT - 8T^{2} \)
3 \( 1 + 0.561iT - 27T^{2} \)
7 \( 1 + 31.0iT - 343T^{2} \)
13 \( 1 - 45.6iT - 2.19e3T^{2} \)
17 \( 1 + 40.4iT - 4.91e3T^{2} \)
19 \( 1 + 91.2T + 6.85e3T^{2} \)
23 \( 1 + 32.2iT - 1.21e4T^{2} \)
29 \( 1 + 35.8T + 2.43e4T^{2} \)
31 \( 1 - 311.T + 2.97e4T^{2} \)
37 \( 1 + 368. iT - 5.06e4T^{2} \)
41 \( 1 + 393.T + 6.89e4T^{2} \)
43 \( 1 - 351. iT - 7.95e4T^{2} \)
47 \( 1 + 230. iT - 1.03e5T^{2} \)
53 \( 1 + 406. iT - 1.48e5T^{2} \)
59 \( 1 - 368.T + 2.05e5T^{2} \)
61 \( 1 + 322.T + 2.26e5T^{2} \)
67 \( 1 - 442. iT - 3.00e5T^{2} \)
71 \( 1 - 667.T + 3.57e5T^{2} \)
73 \( 1 - 84.5iT - 3.89e5T^{2} \)
79 \( 1 - 411.T + 4.93e5T^{2} \)
83 \( 1 - 835. iT - 5.71e5T^{2} \)
89 \( 1 - 799.T + 7.04e5T^{2} \)
97 \( 1 - 768. iT - 9.12e5T^{2} \)
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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

−11.02702119689305835611705154375, −10.34223296999676054283750595817, −9.673303994914895848570034510505, −8.027931237161871424392257441560, −6.98980004613168497416407331010, −6.61284201340044846671770821204, −4.56920389883350203371250224072, −3.73225685174482483524338269278, −2.09739132238555092272592116348, −0.832830105290406522855055924617, 1.87125566795404075754337128300, 3.00890095887306138394614572558, 4.87008820175990593003947564022, 5.88275957735464605953974101622, 6.66123507156817981216777913513, 7.980775333456928087718060525912, 8.583095001214185576224583760089, 9.951403748914494922225129515742, 10.74527572939234908913562845972, 11.94794193790170820643664275543

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