Properties

Label 2-1450-5.4-c1-0-34
Degree $2$
Conductor $1450$
Sign $0.894 + 0.447i$
Analytic cond. $11.5783$
Root an. cond. $3.40269$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s + 3.40i·3-s − 4-s − 3.40·6-s − 0.772i·7-s i·8-s − 8.57·9-s + 1.54·11-s − 3.40i·12-s − 3.85i·13-s + 0.772·14-s + 16-s − 1.22i·17-s − 8.57i·18-s − 6.80·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.96i·3-s − 0.5·4-s − 1.38·6-s − 0.292i·7-s − 0.353i·8-s − 2.85·9-s + 0.466·11-s − 0.982i·12-s − 1.06i·13-s + 0.206·14-s + 0.250·16-s − 0.297i·17-s − 2.02i·18-s − 1.56·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1450\)    =    \(2 \cdot 5^{2} \cdot 29\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(11.5783\)
Root analytic conductor: \(3.40269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1450} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1450,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2176243931\)
\(L(\frac12)\) \(\approx\) \(0.2176243931\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 \)
29 \( 1 + T \)
good3 \( 1 - 3.40iT - 3T^{2} \)
7 \( 1 + 0.772iT - 7T^{2} \)
11 \( 1 - 1.54T + 11T^{2} \)
13 \( 1 + 3.85iT - 13T^{2} \)
17 \( 1 + 1.22iT - 17T^{2} \)
19 \( 1 + 6.80T + 19T^{2} \)
23 \( 1 + 3.40iT - 23T^{2} \)
31 \( 1 + 9.12T + 31T^{2} \)
37 \( 1 + 0.454iT - 37T^{2} \)
41 \( 1 - 3.54T + 41T^{2} \)
43 \( 1 + 3.86iT - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 - 0.318iT - 53T^{2} \)
59 \( 1 + 8.66T + 59T^{2} \)
61 \( 1 + 1.40T + 61T^{2} \)
67 \( 1 - 13.6iT - 67T^{2} \)
71 \( 1 + 13.6T + 71T^{2} \)
73 \( 1 - 9.92iT - 73T^{2} \)
79 \( 1 + 14.8T + 79T^{2} \)
83 \( 1 - 14.8iT - 83T^{2} \)
89 \( 1 - 1.71T + 89T^{2} \)
97 \( 1 - 10.9iT - 97T^{2} \)
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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

−9.342853288062237407826096823811, −8.792854848084977833799238914301, −8.144925291437221285557315551917, −6.95964627326969769802381440585, −5.84134039674881800887001480902, −5.34145005383028637226926434298, −4.28218266764092909919266083303, −3.87590611159362690600602017296, −2.72045447453297021082354482499, −0.084140855267864550246695129046, 1.53314618546455915126020148881, 2.02894207808936407266194157926, 3.15096420029114320446075377747, 4.38980052225456695597415621333, 5.78512556656091652617852662792, 6.31886990202820317296293789613, 7.21690247751807769598771207093, 7.908049929296893064457756334304, 8.923785033516019325799403895069, 9.161688248523255459413092818748

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