Properties

Label 2-145-145.102-c1-0-9
Degree $2$
Conductor $145$
Sign $-0.326 + 0.945i$
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  + (−0.637 − 0.508i)2-s + (0.524 − 2.29i)3-s + (−0.297 − 1.30i)4-s + (1.01 + 1.99i)5-s + (−1.50 + 1.19i)6-s + (3.08 − 1.94i)7-s + (−1.17 + 2.44i)8-s + (−2.30 − 1.10i)9-s + (0.369 − 1.78i)10-s + (−5.09 − 1.78i)11-s − 3.14·12-s + (−0.946 − 0.331i)13-s + (−2.95 − 0.332i)14-s + (5.11 − 1.27i)15-s + (−0.409 + 0.197i)16-s + 4.98i·17-s + ⋯
L(s)  = 1  + (−0.450 − 0.359i)2-s + (0.302 − 1.32i)3-s + (−0.148 − 0.651i)4-s + (0.451 + 0.892i)5-s + (−0.613 + 0.488i)6-s + (1.16 − 0.733i)7-s + (−0.417 + 0.866i)8-s + (−0.766 − 0.369i)9-s + (0.116 − 0.564i)10-s + (−1.53 − 0.537i)11-s − 0.908·12-s + (−0.262 − 0.0918i)13-s + (−0.789 − 0.0889i)14-s + (1.32 − 0.329i)15-s + (−0.102 + 0.0493i)16-s + 1.20i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.326 + 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.326 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $-0.326 + 0.945i$
Analytic conductor: \(1.15783\)
Root analytic conductor: \(1.07602\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (102, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 145,\ (\ :1/2),\ -0.326 + 0.945i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.591338 - 0.829981i\)
\(L(\frac12)\) \(\approx\) \(0.591338 - 0.829981i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.01 - 1.99i)T \)
29 \( 1 + (-1.73 + 5.09i)T \)
good2 \( 1 + (0.637 + 0.508i)T + (0.445 + 1.94i)T^{2} \)
3 \( 1 + (-0.524 + 2.29i)T + (-2.70 - 1.30i)T^{2} \)
7 \( 1 + (-3.08 + 1.94i)T + (3.03 - 6.30i)T^{2} \)
11 \( 1 + (5.09 + 1.78i)T + (8.60 + 6.85i)T^{2} \)
13 \( 1 + (0.946 + 0.331i)T + (10.1 + 8.10i)T^{2} \)
17 \( 1 - 4.98iT - 17T^{2} \)
19 \( 1 + (-2.28 - 1.43i)T + (8.24 + 17.1i)T^{2} \)
23 \( 1 + (-7.13 - 0.804i)T + (22.4 + 5.11i)T^{2} \)
31 \( 1 + (-0.0140 + 0.00158i)T + (30.2 - 6.89i)T^{2} \)
37 \( 1 + (-7.19 - 3.46i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + (4.02 - 4.02i)T - 41iT^{2} \)
43 \( 1 + (3.44 + 4.32i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (7.32 - 3.52i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (0.275 + 2.44i)T + (-51.6 + 11.7i)T^{2} \)
59 \( 1 + 2.07iT - 59T^{2} \)
61 \( 1 + (5.00 - 3.14i)T + (26.4 - 54.9i)T^{2} \)
67 \( 1 + (-4.73 + 1.65i)T + (52.3 - 41.7i)T^{2} \)
71 \( 1 + (-2.57 - 5.35i)T + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (-1.55 + 1.24i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + (9.53 - 3.33i)T + (61.7 - 49.2i)T^{2} \)
83 \( 1 + (-3.62 - 2.27i)T + (36.0 + 74.7i)T^{2} \)
89 \( 1 + (-1.00 - 8.90i)T + (-86.7 + 19.8i)T^{2} \)
97 \( 1 + (-0.259 - 1.13i)T + (-87.3 + 42.0i)T^{2} \)
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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

−13.09663164277904284752804425080, −11.47038918285961917718358548721, −10.76260735889256128318347313367, −9.977000193708124180689685075549, −8.261118709092593969148283703538, −7.70604634015459913696210400019, −6.42148305017010898174856503358, −5.18657159091421287705684427907, −2.65170676657073578920777409124, −1.39446966564039210678247775409, 2.82417566014500440434674040685, 4.84057906637520357002832174476, 5.04516593942623623771610073821, 7.41740743330531910551293494239, 8.475058924477225535688138570529, 9.154988022952961018903871054917, 9.942351850410638911301485384914, 11.25069176354140771867901698126, 12.42823015066944615620618102015, 13.35318791779820700053367724185

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