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Properties

Label 2-145-145.99-c0-0-0
Degree $2$
Conductor $145$
Sign $0.981 - 0.189i$
Analytic cond. $0.0723644$
Root an. cond. $0.269006$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·4-s − i·5-s + i·9-s + (−1 − i)11-s − 16-s + (−1 − i)19-s + 20-s − 25-s + i·29-s + (1 + i)31-s − 36-s + (1 − i)41-s + (1 − i)44-s + 45-s + 49-s + ⋯
L(s)  = 1  + i·4-s − i·5-s + i·9-s + (−1 − i)11-s − 16-s + (−1 − i)19-s + 20-s − 25-s + i·29-s + (1 + i)31-s − 36-s + (1 − i)41-s + (1 − i)44-s + 45-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $0.981 - 0.189i$
Analytic conductor: \(0.0723644\)
Root analytic conductor: \(0.269006\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 145,\ (\ :0),\ 0.981 - 0.189i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6128548394\)
\(L(\frac12)\) \(\approx\) \(0.6128548394\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + iT \)
29 \( 1 - iT \)
good2 \( 1 - iT^{2} \)
3 \( 1 - iT^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (1 + i)T + iT^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + (1 + i)T + iT^{2} \)
23 \( 1 - T^{2} \)
31 \( 1 + (-1 - i)T + iT^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + (-1 + i)T - iT^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + (-1 - i)T + iT^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - 2iT - T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + (1 + i)T + iT^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (1 + i)T + iT^{2} \)
97 \( 1 + iT^{2} \)
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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

−13.19007158956853907013601278393, −12.60653029703028197500961535334, −11.40313183987310958416432208044, −10.47759928528073937459323626242, −8.766585948101750623257275695389, −8.365995397532859272406981295165, −7.21612790594965992491197968021, −5.46047728948515655924322566353, −4.37461038896300874501784182487, −2.66446834598452832199469494415, 2.37441829380017958106060039538, 4.23963785046997912056196187044, 5.85627077927153378918702552554, 6.66425983048885149547281387292, 7.929057424431366116466128070148, 9.608335148010529482036674461967, 10.15441551610510815076338042987, 11.10773304512960009381429154755, 12.24017337137894231430027189121, 13.42152706905986775804239387810

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