Properties

Label 2-145-5.4-c1-0-1
Degree $2$
Conductor $145$
Sign $-0.977 - 0.210i$
Analytic cond. $1.15783$
Root an. cond. $1.07602$
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  + 1.73i·2-s + 2.52i·3-s − 0.999·4-s + (−2.18 − 0.469i)5-s − 4.37·6-s − 1.58i·7-s + 1.73i·8-s − 3.37·9-s + (0.813 − 3.78i)10-s + 6.37·11-s − 2.52i·12-s − 0.939i·13-s + 2.74·14-s + (1.18 − 5.51i)15-s − 5·16-s + 5.04i·17-s + ⋯
L(s)  = 1  + 1.22i·2-s + 1.45i·3-s − 0.499·4-s + (−0.977 − 0.210i)5-s − 1.78·6-s − 0.598i·7-s + 0.612i·8-s − 1.12·9-s + (0.257 − 1.19i)10-s + 1.92·11-s − 0.728i·12-s − 0.260i·13-s + 0.733·14-s + (0.306 − 1.42i)15-s − 1.25·16-s + 1.22i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.210i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 - 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $-0.977 - 0.210i$
Analytic conductor: \(1.15783\)
Root analytic conductor: \(1.07602\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 145,\ (\ :1/2),\ -0.977 - 0.210i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.109610 + 1.03157i\)
\(L(\frac12)\) \(\approx\) \(0.109610 + 1.03157i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.18 + 0.469i)T \)
29 \( 1 - T \)
good2 \( 1 - 1.73iT - 2T^{2} \)
3 \( 1 - 2.52iT - 3T^{2} \)
7 \( 1 + 1.58iT - 7T^{2} \)
11 \( 1 - 6.37T + 11T^{2} \)
13 \( 1 + 0.939iT - 13T^{2} \)
17 \( 1 - 5.04iT - 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + 3.46iT - 23T^{2} \)
31 \( 1 - 2.37T + 31T^{2} \)
37 \( 1 + 10.0iT - 37T^{2} \)
41 \( 1 - 6.74T + 41T^{2} \)
43 \( 1 + 5.69iT - 43T^{2} \)
47 \( 1 - 5.69iT - 47T^{2} \)
53 \( 1 + 0.939iT - 53T^{2} \)
59 \( 1 + 0.744T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 + 8.51iT - 67T^{2} \)
71 \( 1 + 4.74T + 71T^{2} \)
73 \( 1 - 6.92iT - 73T^{2} \)
79 \( 1 + 5.62T + 79T^{2} \)
83 \( 1 + 16.7iT - 83T^{2} \)
89 \( 1 + 10.7T + 89T^{2} \)
97 \( 1 + 6.92iT - 97T^{2} \)
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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

−14.31874399434762927045455261166, −12.50939259847363121809259103225, −11.28020141983659311769453636907, −10.54125883439026440307578483747, −9.095691124682337976065135095210, −8.421934579477768410399368281305, −7.11927841538927821800191863085, −6.00791994071991985641030865922, −4.41627532927648895385927444905, −3.91669487215032119232006392549, 1.22484106305275691711494156090, 2.71425187992633401814744734436, 4.14871669065862588100463645526, 6.47571411923543533438776391264, 7.10997106416035410010324836321, 8.493721109413056414324877490466, 9.558636311981072011114231636562, 11.21285709905194519340110771450, 11.89722531616134555691754174748, 12.11032881635976759245030988265

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