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Properties

Label 2-145-5.4-c1-0-0
Degree $2$
Conductor $145$
Sign $-0.878 + 0.478i$
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  + 2.77i·2-s − 0.269i·3-s − 5.67·4-s + (−1.96 + 1.06i)5-s + 0.747·6-s + 1.86i·7-s − 10.1i·8-s + 2.92·9-s + (−2.96 − 5.43i)10-s − 3.25·11-s + 1.53i·12-s + 3.40i·13-s − 5.17·14-s + (0.288 + 0.529i)15-s + 16.8·16-s + 2.40i·17-s + ⋯
L(s)  = 1  + 1.95i·2-s − 0.155i·3-s − 2.83·4-s + (−0.878 + 0.478i)5-s + 0.305·6-s + 0.706i·7-s − 3.59i·8-s + 0.975·9-s + (−0.937 − 1.72i)10-s − 0.980·11-s + 0.442i·12-s + 0.943i·13-s − 1.38·14-s + (0.0745 + 0.136i)15-s + 4.21·16-s + 0.584i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.878 + 0.478i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.878 + 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $-0.878 + 0.478i$
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: \((2,\ 145,\ (\ :1/2),\ -0.878 + 0.478i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.176681 - 0.693620i\)
\(L(\frac12)\) \(\approx\) \(0.176681 - 0.693620i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.96 - 1.06i)T \)
29 \( 1 - T \)
good2 \( 1 - 2.77iT - 2T^{2} \)
3 \( 1 + 0.269iT - 3T^{2} \)
7 \( 1 - 1.86iT - 7T^{2} \)
11 \( 1 + 3.25T + 11T^{2} \)
13 \( 1 - 3.40iT - 13T^{2} \)
17 \( 1 - 2.40iT - 17T^{2} \)
19 \( 1 - 0.674T + 19T^{2} \)
23 \( 1 - 7.41iT - 23T^{2} \)
31 \( 1 - 5.25T + 31T^{2} \)
37 \( 1 + 1.86iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 3.46iT - 43T^{2} \)
47 \( 1 + 4.00iT - 47T^{2} \)
53 \( 1 - 0.877iT - 53T^{2} \)
59 \( 1 + 10T + 59T^{2} \)
61 \( 1 - 11.8T + 61T^{2} \)
67 \( 1 + 7.95iT - 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + 0.607iT - 73T^{2} \)
79 \( 1 + 8.60T + 79T^{2} \)
83 \( 1 + 2.40iT - 83T^{2} \)
89 \( 1 - 8.50T + 89T^{2} \)
97 \( 1 - 13.1iT - 97T^{2} \)
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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

−13.92301962737694538172537577965, −13.04352726521785670563859635791, −11.99102528716145205147363537132, −10.25090962241174288617134187493, −9.110253611982722406612870267419, −8.013616752869046414211176127195, −7.32850132372482763009875020732, −6.36784564754320836179820049315, −5.08835621184021306723856791168, −3.89565328243853922571192153020, 0.77987242225437211767586115719, 2.91362387132760813110666347490, 4.22057669263594512958222748087, 4.95869877159582691024536261954, 7.66375450998325171366980439651, 8.591927905541423311622636646840, 9.967340911458833811408258902947, 10.48658769683817806905079163029, 11.42851646343676511139893209364, 12.57662167923524132921377691636

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