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Properties

Label 2-152-8.5-c1-0-6
Degree $2$
Conductor $152$
Sign $0.552 - 0.833i$
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  + (0.456 + 1.33i)2-s − 2.09i·3-s + (−1.58 + 1.22i)4-s + 3.36i·5-s + (2.80 − 0.956i)6-s + 4.47·7-s + (−2.35 − 1.56i)8-s − 1.39·9-s + (−4.50 + 1.53i)10-s + 0.608i·11-s + (2.56 + 3.31i)12-s − 1.03i·13-s + (2.04 + 5.98i)14-s + 7.05·15-s + (1.01 − 3.86i)16-s − 3.06·17-s + ⋯
L(s)  = 1  + (0.322 + 0.946i)2-s − 1.20i·3-s + (−0.791 + 0.610i)4-s + 1.50i·5-s + (1.14 − 0.390i)6-s + 1.68·7-s + (−0.833 − 0.552i)8-s − 0.463·9-s + (−1.42 + 0.486i)10-s + 0.183i·11-s + (0.739 + 0.957i)12-s − 0.288i·13-s + (0.545 + 1.59i)14-s + 1.82·15-s + (0.253 − 0.967i)16-s − 0.743·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.552 - 0.833i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.552 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $0.552 - 0.833i$
Analytic conductor: \(1.21372\)
Root analytic conductor: \(1.10169\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 152,\ (\ :1/2),\ 0.552 - 0.833i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17378 + 0.630568i\)
\(L(\frac12)\) \(\approx\) \(1.17378 + 0.630568i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.456 - 1.33i)T \)
19 \( 1 + iT \)
good3 \( 1 + 2.09iT - 3T^{2} \)
5 \( 1 - 3.36iT - 5T^{2} \)
7 \( 1 - 4.47T + 7T^{2} \)
11 \( 1 - 0.608iT - 11T^{2} \)
13 \( 1 + 1.03iT - 13T^{2} \)
17 \( 1 + 3.06T + 17T^{2} \)
23 \( 1 + 8.50T + 23T^{2} \)
29 \( 1 + 7.27iT - 29T^{2} \)
31 \( 1 - 4.02T + 31T^{2} \)
37 \( 1 - 4.31iT - 37T^{2} \)
41 \( 1 + 4.15T + 41T^{2} \)
43 \( 1 + 6.27iT - 43T^{2} \)
47 \( 1 - 4.73T + 47T^{2} \)
53 \( 1 - 6.98iT - 53T^{2} \)
59 \( 1 + 2.64iT - 59T^{2} \)
61 \( 1 - 5.11iT - 61T^{2} \)
67 \( 1 + 2.62iT - 67T^{2} \)
71 \( 1 - 12.0T + 71T^{2} \)
73 \( 1 + 12.5T + 73T^{2} \)
79 \( 1 + 0.913T + 79T^{2} \)
83 \( 1 - 0.887iT - 83T^{2} \)
89 \( 1 + 7.61T + 89T^{2} \)
97 \( 1 + 5.93T + 97T^{2} \)
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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

−13.76061751786548514496323960627, −12.18169587498596429638835207917, −11.45766253097143760982583531290, −10.17935961043155450737217953814, −8.312479572866939267949483442744, −7.69078327862456925370946978606, −6.86622585214902461954342256587, −5.92178838595648117535081549406, −4.30584053388723442114280554108, −2.30519157334385807502225078658, 1.66859414021996525581259853281, 4.08820969149664341881263681944, 4.68781424445195681441903770199, 5.47685915271131768664864908469, 8.253601566768449939070644681555, 8.875333602715582438451919624418, 9.890652515573930060390676082807, 10.91526066518763283989563608840, 11.71910784398227220778170685924, 12.62406445276576370905197452204

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