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Properties

Label 2-296-296.101-c1-0-3
Degree $2$
Conductor $296$
Sign $0.830 - 0.556i$
Analytic cond. $2.36357$
Root an. cond. $1.53739$
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  + (0.254 − 1.39i)2-s + (−1.20 + 0.696i)3-s + (−1.87 − 0.707i)4-s + (1.27 + 2.20i)5-s + (0.662 + 1.85i)6-s + (−0.577 − 1.00i)7-s + (−1.45 + 2.42i)8-s + (−0.530 + 0.919i)9-s + (3.38 − 1.20i)10-s + 1.24i·11-s + (2.74 − 0.449i)12-s + (2.50 + 4.34i)13-s + (−1.53 + 0.549i)14-s + (−3.06 − 1.76i)15-s + (3.00 + 2.64i)16-s + (0.690 + 0.398i)17-s + ⋯
L(s)  = 1  + (0.179 − 0.983i)2-s + (−0.696 + 0.401i)3-s + (−0.935 − 0.353i)4-s + (0.568 + 0.984i)5-s + (0.270 + 0.757i)6-s + (−0.218 − 0.378i)7-s + (−0.515 + 0.856i)8-s + (−0.176 + 0.306i)9-s + (1.07 − 0.382i)10-s + 0.376i·11-s + (0.793 − 0.129i)12-s + (0.695 + 1.20i)13-s + (−0.411 + 0.146i)14-s + (−0.791 − 0.456i)15-s + (0.750 + 0.661i)16-s + (0.167 + 0.0967i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 - 0.556i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.830 - 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(296\)    =    \(2^{3} \cdot 37\)
Sign: $0.830 - 0.556i$
Analytic conductor: \(2.36357\)
Root analytic conductor: \(1.53739\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{296} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 296,\ (\ :1/2),\ 0.830 - 0.556i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.923572 + 0.280735i\)
\(L(\frac12)\) \(\approx\) \(0.923572 + 0.280735i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.254 + 1.39i)T \)
37 \( 1 + (3.76 - 4.77i)T \)
good3 \( 1 + (1.20 - 0.696i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.27 - 2.20i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.577 + 1.00i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 1.24iT - 11T^{2} \)
13 \( 1 + (-2.50 - 4.34i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.690 - 0.398i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.35 - 4.07i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 0.721iT - 23T^{2} \)
29 \( 1 + 3.82T + 29T^{2} \)
31 \( 1 + 2.61iT - 31T^{2} \)
41 \( 1 + (-4.92 - 8.53i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 3.41T + 43T^{2} \)
47 \( 1 + 6.38T + 47T^{2} \)
53 \( 1 + (4.12 + 2.37i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.69 - 2.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.89 + 5.02i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.35 + 4.24i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.82 - 4.88i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 16.2T + 73T^{2} \)
79 \( 1 + (7.63 - 4.41i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.05 - 4.07i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.50 - 0.867i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 11.3iT - 97T^{2} \)
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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

−11.45654306021328209723925567372, −11.07668773655868460828617331973, −10.07690328476801881171156543524, −9.657414728406044527214472615822, −8.196303201399662771104097129698, −6.64881307619801486869622596981, −5.75888557077159100374584866399, −4.55393878915397614983181485054, −3.39158288106484045770145024219, −1.91159428909672480542718897202, 0.77668098902245512410539026083, 3.42219767443799331467602705522, 5.19535831504355408602922003342, 5.60653801962189719855804865025, 6.52270152636492426427467932761, 7.72233101968375173423991371602, 8.868580516382868048229364540429, 9.332824967170756131824341729150, 10.78849197272744766998855289595, 12.02605841196063239615247029216

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