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Properties

Label 2-296-296.19-c1-0-19
Degree $2$
Conductor $296$
Sign $0.981 - 0.193i$
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  + (0.0425 + 1.41i)2-s + (−0.111 − 0.132i)3-s + (−1.99 + 0.120i)4-s + (1.52 − 0.711i)5-s + (0.182 − 0.162i)6-s + (1.41 − 3.89i)7-s + (−0.255 − 2.81i)8-s + (0.515 − 2.92i)9-s + (1.07 + 2.12i)10-s + (3.55 + 2.05i)11-s + (0.238 + 0.251i)12-s + (−2.96 − 2.07i)13-s + (5.56 + 1.83i)14-s + (−0.264 − 0.123i)15-s + (3.97 − 0.480i)16-s + (−6.54 + 4.58i)17-s + ⋯
L(s)  = 1  + (0.0300 + 0.999i)2-s + (−0.0642 − 0.0765i)3-s + (−0.998 + 0.0601i)4-s + (0.682 − 0.318i)5-s + (0.0746 − 0.0665i)6-s + (0.535 − 1.47i)7-s + (−0.0901 − 0.995i)8-s + (0.171 − 0.974i)9-s + (0.338 + 0.672i)10-s + (1.07 + 0.619i)11-s + (0.0687 + 0.0725i)12-s + (−0.821 − 0.574i)13-s + (1.48 + 0.491i)14-s + (−0.0682 − 0.0318i)15-s + (0.992 − 0.120i)16-s + (−1.58 + 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.193i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 - 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(296\)    =    \(2^{3} \cdot 37\)
Sign: $0.981 - 0.193i$
Analytic conductor: \(2.36357\)
Root analytic conductor: \(1.53739\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{296} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 296,\ (\ :1/2),\ 0.981 - 0.193i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34308 + 0.131376i\)
\(L(\frac12)\) \(\approx\) \(1.34308 + 0.131376i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0425 - 1.41i)T \)
37 \( 1 + (-2.39 + 5.59i)T \)
good3 \( 1 + (0.111 + 0.132i)T + (-0.520 + 2.95i)T^{2} \)
5 \( 1 + (-1.52 + 0.711i)T + (3.21 - 3.83i)T^{2} \)
7 \( 1 + (-1.41 + 3.89i)T + (-5.36 - 4.49i)T^{2} \)
11 \( 1 + (-3.55 - 2.05i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.96 + 2.07i)T + (4.44 + 12.2i)T^{2} \)
17 \( 1 + (6.54 - 4.58i)T + (5.81 - 15.9i)T^{2} \)
19 \( 1 + (-4.70 - 0.411i)T + (18.7 + 3.29i)T^{2} \)
23 \( 1 + (-3.03 - 0.812i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-3.69 + 0.990i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (1.65 + 1.65i)T + 31iT^{2} \)
41 \( 1 + (-0.905 + 0.159i)T + (38.5 - 14.0i)T^{2} \)
43 \( 1 + (-2.46 - 2.46i)T + 43iT^{2} \)
47 \( 1 + (5.70 - 3.29i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.89 - 5.19i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (-6.37 - 2.97i)T + (37.9 + 45.1i)T^{2} \)
61 \( 1 + (6.47 - 9.24i)T + (-20.8 - 57.3i)T^{2} \)
67 \( 1 + (3.54 - 9.73i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (1.54 + 1.84i)T + (-12.3 + 69.9i)T^{2} \)
73 \( 1 + 8.62iT - 73T^{2} \)
79 \( 1 + (-11.8 + 5.50i)T + (50.7 - 60.5i)T^{2} \)
83 \( 1 + (-1.96 + 11.1i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (-0.638 + 1.36i)T + (-57.2 - 68.1i)T^{2} \)
97 \( 1 + (4.86 - 18.1i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.99368799027188484611985022394, −10.62351691067143911705953440769, −9.631081586699782231699678452430, −9.035101110829373669429930561421, −7.64093492412785418968818536429, −6.95685792281562003479046154630, −6.02588720127305756181720603376, −4.67226313143922098768203571748, −3.88984610678517154847181387762, −1.17957546456998888255107380700, 1.94688685019996499209921338580, 2.77917691408131199353064287503, 4.66785050348934200882493835168, 5.35688038477284904733113469471, 6.71187031219749673367639562753, 8.330016205536440387359260978515, 9.175743730564852340392777293491, 9.766422628279591021837191428603, 11.13739234811686532989684270168, 11.53538096479975809796680739433

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