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Properties

Label 2-145-29.28-c1-0-3
Degree $2$
Conductor $145$
Sign $0.0556 - 0.998i$
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  + 1.30i·2-s + 0.537i·3-s + 0.299·4-s + 5-s − 0.700·6-s − 1.29·7-s + 2.99i·8-s + 2.71·9-s + 1.30i·10-s − 0.537i·11-s + 0.160i·12-s − 5.71·13-s − 1.69i·14-s + 0.537i·15-s − 3.31·16-s − 6.91i·17-s + ⋯
L(s)  = 1  + 0.922i·2-s + 0.310i·3-s + 0.149·4-s + 0.447·5-s − 0.285·6-s − 0.491·7-s + 1.06i·8-s + 0.903·9-s + 0.412i·10-s − 0.161i·11-s + 0.0464i·12-s − 1.58·13-s − 0.452i·14-s + 0.138i·15-s − 0.827·16-s − 1.67i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0556 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0556 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $0.0556 - 0.998i$
Analytic conductor: \(1.15783\)
Root analytic conductor: \(1.07602\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (86, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 145,\ (\ :1/2),\ 0.0556 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.916560 + 0.866900i\)
\(L(\frac12)\) \(\approx\) \(0.916560 + 0.866900i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
29 \( 1 + (-0.299 + 5.37i)T \)
good2 \( 1 - 1.30iT - 2T^{2} \)
3 \( 1 - 0.537iT - 3T^{2} \)
7 \( 1 + 1.29T + 7T^{2} \)
11 \( 1 + 0.537iT - 11T^{2} \)
13 \( 1 + 5.71T + 13T^{2} \)
17 \( 1 + 6.91iT - 17T^{2} \)
19 \( 1 - 4.30iT - 19T^{2} \)
23 \( 1 - 5.01T + 23T^{2} \)
31 \( 1 + 8.44iT - 31T^{2} \)
37 \( 1 + 7.98iT - 37T^{2} \)
41 \( 1 + 0.698iT - 41T^{2} \)
43 \( 1 - 10.2iT - 43T^{2} \)
47 \( 1 + 0.537iT - 47T^{2} \)
53 \( 1 - 7.11T + 53T^{2} \)
59 \( 1 + 7.71T + 59T^{2} \)
61 \( 1 - 5.91iT - 61T^{2} \)
67 \( 1 + 9.01T + 67T^{2} \)
71 \( 1 + 4.28T + 71T^{2} \)
73 \( 1 - 6.21iT - 73T^{2} \)
79 \( 1 - 10.2iT - 79T^{2} \)
83 \( 1 + 2.70T + 83T^{2} \)
89 \( 1 - 9.67iT - 89T^{2} \)
97 \( 1 + 2.07iT - 97T^{2} \)
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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

−13.52009534586050623017846878307, −12.41018957565350608735111252967, −11.31273117567351878626936998791, −9.960339939109492255377409886476, −9.354268791870579598614692943461, −7.66425740271153784813463286524, −7.02240022142731459812814913238, −5.77454810245506480204364415517, −4.65466071700818048037903256614, −2.57949465807300233005271469073, 1.69718507159421331233031470623, 3.10748322824270907517974505288, 4.76583666621125208548521017164, 6.58236105731666068803614645463, 7.25194686903495725995506576691, 9.035105806642557997769587241157, 10.12079407181254602969244465614, 10.61572978638732808332179461419, 12.08343053442731732988021798043, 12.67462688514460083704161215431

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