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Properties

Label 2-152-19.11-c1-0-1
Degree $2$
Conductor $152$
Sign $-0.0977 - 0.995i$
Analytic cond. $1.21372$
Root an. cond. $1.10169$
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  + (−0.5 + 0.866i)3-s + (−1.5 + 2.59i)5-s + (1 + 1.73i)9-s − 4·11-s + (2.5 + 4.33i)13-s + (−1.5 − 2.59i)15-s + (2.5 − 4.33i)17-s + (4 − 1.73i)19-s + (0.5 + 0.866i)23-s + (−2 − 3.46i)25-s − 5·27-s + (−1.5 − 2.59i)29-s + 4·31-s + (2 − 3.46i)33-s + 2·37-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.670 + 1.16i)5-s + (0.333 + 0.577i)9-s − 1.20·11-s + (0.693 + 1.20i)13-s + (−0.387 − 0.670i)15-s + (0.606 − 1.05i)17-s + (0.917 − 0.397i)19-s + (0.104 + 0.180i)23-s + (−0.400 − 0.692i)25-s − 0.962·27-s + (−0.278 − 0.482i)29-s + 0.718·31-s + (0.348 − 0.603i)33-s + 0.328·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $-0.0977 - 0.995i$
Analytic conductor: \(1.21372\)
Root analytic conductor: \(1.10169\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 152,\ (\ :1/2),\ -0.0977 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.593257 + 0.654372i\)
\(L(\frac12)\) \(\approx\) \(0.593257 + 0.654372i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (-4 + 1.73i)T \)
good3 \( 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + (-2.5 - 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.5 + 4.33i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (-2.5 + 4.33i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.5 + 9.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.5 - 4.33i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.5 - 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.5 - 11.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.5 - 4.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.5 + 7.79i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.5 - 14.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 16T + 83T^{2} \)
89 \( 1 + (1.5 + 2.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.5 + 11.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.49623060698242654792520311593, −11.92785765795071117290117393839, −11.13253141962830958834015744702, −10.46177320669096020045510745729, −9.367134985478709320882469849148, −7.75348182049944111759256174068, −7.09112036315099228024375911223, −5.53051998006997938078980694943, −4.22027592841996732724125269361, −2.78405572274053258556836322948, 0.979914450721557854737958164682, 3.50875417154481424874690094537, 5.03494778594923827488258126283, 6.09566197658560200325766043446, 7.78111380016531325454854692236, 8.216548688067344586752233555432, 9.656916818908247081094457171789, 10.80376756342128163847906372351, 12.01255057718805074530326390348, 12.78901816022090662434005503767

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