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Properties

Label 2-145-145.144-c1-0-8
Degree $2$
Conductor $145$
Sign $0.998 + 0.0492i$
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  + 2-s + 1.56·3-s − 4-s + (1.78 − 1.35i)5-s + 1.56·6-s + 3.46i·7-s − 3·8-s − 0.561·9-s + (1.78 − 1.35i)10-s − 6.16i·11-s − 1.56·12-s + 2.70i·13-s + 3.46i·14-s + (2.78 − 2.11i)15-s − 16-s − 2·17-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.901·3-s − 0.5·4-s + (0.796 − 0.604i)5-s + 0.637·6-s + 1.30i·7-s − 1.06·8-s − 0.187·9-s + (0.563 − 0.427i)10-s − 1.85i·11-s − 0.450·12-s + 0.750i·13-s + 0.925i·14-s + (0.717 − 0.545i)15-s − 0.250·16-s − 0.485·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0492i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0492i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $0.998 + 0.0492i$
Analytic conductor: \(1.15783\)
Root analytic conductor: \(1.07602\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (144, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 145,\ (\ :1/2),\ 0.998 + 0.0492i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75806 - 0.0433100i\)
\(L(\frac12)\) \(\approx\) \(1.75806 - 0.0433100i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.78 + 1.35i)T \)
29 \( 1 + (4.12 + 3.46i)T \)
good2 \( 1 - T + 2T^{2} \)
3 \( 1 - 1.56T + 3T^{2} \)
7 \( 1 - 3.46iT - 7T^{2} \)
11 \( 1 + 6.16iT - 11T^{2} \)
13 \( 1 - 2.70iT - 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 - 3.46iT - 23T^{2} \)
31 \( 1 - 0.759iT - 31T^{2} \)
37 \( 1 + 1.12T + 37T^{2} \)
41 \( 1 + 5.40iT - 41T^{2} \)
43 \( 1 - 9.56T + 43T^{2} \)
47 \( 1 - 0.684T + 47T^{2} \)
53 \( 1 + 2.70iT - 53T^{2} \)
59 \( 1 - 7.12T + 59T^{2} \)
61 \( 1 - 12.3iT - 61T^{2} \)
67 \( 1 - 1.94iT - 67T^{2} \)
71 \( 1 - 14.2T + 71T^{2} \)
73 \( 1 - 1.12T + 73T^{2} \)
79 \( 1 + 7.68iT - 79T^{2} \)
83 \( 1 - 10.3iT - 83T^{2} \)
89 \( 1 + 5.40iT - 89T^{2} \)
97 \( 1 + 8.24T + 97T^{2} \)
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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

−13.43889011164588289232789730080, −12.35153613720270175233959531915, −11.37213743269008301001812661282, −9.501735370232943818380554372661, −8.864759628251849098009506582360, −8.374560619853472157507830979116, −5.93600880673022359914214235589, −5.53467189363769808939408089058, −3.77798468214998008915571869237, −2.46376395633078951884282749189, 2.52846762786872877591541127261, 3.88331016351495229100794981424, 5.06138034405544140203796528198, 6.65864231834549312178151051258, 7.69366963351013351665521093146, 9.155103672950530905846552952936, 9.895512590086166173701038447378, 10.93418588853145970305628468137, 12.66577507342342333859243770239, 13.26535324954225000644954924716

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