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Properties

Label 2-145-5.4-c1-0-6
Degree $2$
Conductor $145$
Sign $0.575 - 0.817i$
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  + 0.156i·2-s + 2.56i·3-s + 1.97·4-s + (1.28 − 1.82i)5-s − 0.401·6-s − 1.09i·7-s + 0.623i·8-s − 3.57·9-s + (0.286 + 0.201i)10-s − 4.40·11-s + 5.06i·12-s + 3.97i·13-s + 0.171·14-s + (4.68 + 3.29i)15-s + 3.85·16-s − 6.22i·17-s + ⋯
L(s)  = 1  + 0.110i·2-s + 1.48i·3-s + 0.987·4-s + (0.575 − 0.817i)5-s − 0.164·6-s − 0.413i·7-s + 0.220i·8-s − 1.19·9-s + (0.0906 + 0.0637i)10-s − 1.32·11-s + 1.46i·12-s + 1.10i·13-s + 0.0458·14-s + (1.21 + 0.851i)15-s + 0.963·16-s − 1.50i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.575 - 0.817i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.575 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $0.575 - 0.817i$
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: \((2,\ 145,\ (\ :1/2),\ 0.575 - 0.817i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19485 + 0.620273i\)
\(L(\frac12)\) \(\approx\) \(1.19485 + 0.620273i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.28 + 1.82i)T \)
29 \( 1 - T \)
good2 \( 1 - 0.156iT - 2T^{2} \)
3 \( 1 - 2.56iT - 3T^{2} \)
7 \( 1 + 1.09iT - 7T^{2} \)
11 \( 1 + 4.40T + 11T^{2} \)
13 \( 1 - 3.97iT - 13T^{2} \)
17 \( 1 + 6.22iT - 17T^{2} \)
19 \( 1 + 6.97T + 19T^{2} \)
23 \( 1 + 0.780iT - 23T^{2} \)
31 \( 1 - 6.40T + 31T^{2} \)
37 \( 1 - 1.09iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 0.376iT - 43T^{2} \)
47 \( 1 - 4.75iT - 47T^{2} \)
53 \( 1 + 11.2iT - 53T^{2} \)
59 \( 1 + 10T + 59T^{2} \)
61 \( 1 + 1.14T + 61T^{2} \)
67 \( 1 - 5.90iT - 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 - 8.72iT - 73T^{2} \)
79 \( 1 - 5.54T + 79T^{2} \)
83 \( 1 - 6.22iT - 83T^{2} \)
89 \( 1 - 10.8T + 89T^{2} \)
97 \( 1 - 17.4iT - 97T^{2} \)
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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

−13.33398270687392890820488891156, −12.05401848290328304994721366999, −10.94148642932433863704871027291, −10.23393874058491752016333589198, −9.342434829249614183120278973942, −8.170860650825783090267545135127, −6.64669000630911676622173170837, −5.27257905593781178201918173056, −4.37802160209999028080511191462, −2.53360146766783568950618752460, 1.97666220818795903276714262553, 2.85443478300642238834017190186, 5.84988259042812221546919245826, 6.38287631687183990141387839303, 7.55955201289859036131222780488, 8.270919659415120463685346744258, 10.40653562983824285032945388700, 10.76007485631867381154879486996, 12.22099335639164129237716150611, 12.81423063081128500323036059973

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