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Properties

Label 2-145-145.47-c1-0-8
Degree $2$
Conductor $145$
Sign $0.931 - 0.364i$
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  + (−0.419 + 0.871i)2-s + (1.77 − 2.22i)3-s + (0.663 + 0.832i)4-s + (0.626 + 2.14i)5-s + (1.19 + 2.47i)6-s + (−1.38 + 0.156i)7-s + (−2.88 + 0.659i)8-s + (−1.12 − 4.93i)9-s + (−2.13 − 0.354i)10-s + (4.68 − 2.94i)11-s + 3.02·12-s + (−3.87 + 2.43i)13-s + (0.446 − 1.27i)14-s + (5.87 + 2.40i)15-s + (0.164 − 0.719i)16-s − 5.34i·17-s + ⋯
L(s)  = 1  + (−0.296 + 0.616i)2-s + (1.02 − 1.28i)3-s + (0.331 + 0.416i)4-s + (0.280 + 0.959i)5-s + (0.486 + 1.01i)6-s + (−0.524 + 0.0591i)7-s + (−1.02 + 0.233i)8-s + (−0.375 − 1.64i)9-s + (−0.674 − 0.112i)10-s + (1.41 − 0.886i)11-s + 0.872·12-s + (−1.07 + 0.675i)13-s + (0.119 − 0.341i)14-s + (1.51 + 0.621i)15-s + (0.0410 − 0.179i)16-s − 1.29i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.931 - 0.364i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.931 - 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $0.931 - 0.364i$
Analytic conductor: \(1.15783\)
Root analytic conductor: \(1.07602\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 145,\ (\ :1/2),\ 0.931 - 0.364i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31974 + 0.248801i\)
\(L(\frac12)\) \(\approx\) \(1.31974 + 0.248801i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.626 - 2.14i)T \)
29 \( 1 + (-3.20 + 4.32i)T \)
good2 \( 1 + (0.419 - 0.871i)T + (-1.24 - 1.56i)T^{2} \)
3 \( 1 + (-1.77 + 2.22i)T + (-0.667 - 2.92i)T^{2} \)
7 \( 1 + (1.38 - 0.156i)T + (6.82 - 1.55i)T^{2} \)
11 \( 1 + (-4.68 + 2.94i)T + (4.77 - 9.91i)T^{2} \)
13 \( 1 + (3.87 - 2.43i)T + (5.64 - 11.7i)T^{2} \)
17 \( 1 + 5.34iT - 17T^{2} \)
19 \( 1 + (3.75 + 0.423i)T + (18.5 + 4.22i)T^{2} \)
23 \( 1 + (0.812 - 2.32i)T + (-17.9 - 14.3i)T^{2} \)
31 \( 1 + (-1.32 - 3.78i)T + (-24.2 + 19.3i)T^{2} \)
37 \( 1 + (0.826 + 3.61i)T + (-33.3 + 16.0i)T^{2} \)
41 \( 1 + (5.71 - 5.71i)T - 41iT^{2} \)
43 \( 1 + (-0.669 + 0.322i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (0.00613 - 0.0268i)T + (-42.3 - 20.3i)T^{2} \)
53 \( 1 + (1.64 - 0.574i)T + (41.4 - 33.0i)T^{2} \)
59 \( 1 - 4.00iT - 59T^{2} \)
61 \( 1 + (-3.92 + 0.442i)T + (59.4 - 13.5i)T^{2} \)
67 \( 1 + (-6.04 - 3.79i)T + (29.0 + 60.3i)T^{2} \)
71 \( 1 + (-3.42 - 0.781i)T + (63.9 + 30.8i)T^{2} \)
73 \( 1 + (-1.39 - 2.88i)T + (-45.5 + 57.0i)T^{2} \)
79 \( 1 + (7.95 + 5.00i)T + (34.2 + 71.1i)T^{2} \)
83 \( 1 + (-14.7 - 1.66i)T + (80.9 + 18.4i)T^{2} \)
89 \( 1 + (-12.9 + 4.52i)T + (69.5 - 55.4i)T^{2} \)
97 \( 1 + (-5.48 - 6.87i)T + (-21.5 + 94.5i)T^{2} \)
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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

−13.39580578316704090859433229623, −12.12799522012690987250654728158, −11.52183319961159542756770058050, −9.585809158458509999735550375510, −8.780704600226107642801394808546, −7.63615404885467116135633083300, −6.76470562975508158298889533655, −6.39013938079769979611091400137, −3.35693737267165175250649836710, −2.35906536742486683780491039617, 2.07186430621745110753848121104, 3.67992502273573403014391451257, 4.84482876834715088989897662262, 6.44001070517964518530492649645, 8.369637152932274450637053504763, 9.239575839817342805201212442087, 9.910201368743937843253300997094, 10.47037835455409958088238869786, 12.07090584123991557435846920627, 12.80678242384960759546451451548

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