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Properties

Label 2-296-296.101-c1-0-21
Degree $2$
Conductor $296$
Sign $-0.901 + 0.431i$
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  + (−1.17 − 0.781i)2-s + (−0.593 + 0.342i)3-s + (0.778 + 1.84i)4-s + (−1.02 − 1.77i)5-s + (0.967 + 0.0599i)6-s + (0.355 + 0.615i)7-s + (0.521 − 2.77i)8-s + (−1.26 + 2.19i)9-s + (−0.179 + 2.89i)10-s − 5.64i·11-s + (−1.09 − 0.826i)12-s + (0.935 + 1.62i)13-s + (0.0621 − 1.00i)14-s + (1.21 + 0.703i)15-s + (−2.78 + 2.86i)16-s + (−5.77 − 3.33i)17-s + ⋯
L(s)  = 1  + (−0.833 − 0.552i)2-s + (−0.342 + 0.197i)3-s + (0.389 + 0.921i)4-s + (−0.458 − 0.794i)5-s + (0.395 + 0.0244i)6-s + (0.134 + 0.232i)7-s + (0.184 − 0.982i)8-s + (−0.421 + 0.730i)9-s + (−0.0567 + 0.915i)10-s − 1.70i·11-s + (−0.315 − 0.238i)12-s + (0.259 + 0.449i)13-s + (0.0166 − 0.268i)14-s + (0.314 + 0.181i)15-s + (−0.696 + 0.717i)16-s + (−1.39 − 0.807i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.901 + 0.431i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.901 + 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(296\)    =    \(2^{3} \cdot 37\)
Sign: $-0.901 + 0.431i$
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: \((2,\ 296,\ (\ :1/2),\ -0.901 + 0.431i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0816881 - 0.359674i\)
\(L(\frac12)\) \(\approx\) \(0.0816881 - 0.359674i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.17 + 0.781i)T \)
37 \( 1 + (5.59 - 2.38i)T \)
good3 \( 1 + (0.593 - 0.342i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.02 + 1.77i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.355 - 0.615i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 5.64iT - 11T^{2} \)
13 \( 1 + (-0.935 - 1.62i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (5.77 + 3.33i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.49 + 4.32i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 2.13iT - 23T^{2} \)
29 \( 1 + 9.00T + 29T^{2} \)
31 \( 1 + 6.96iT - 31T^{2} \)
41 \( 1 + (-3.42 - 5.93i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 6.29T + 43T^{2} \)
47 \( 1 + 0.475T + 47T^{2} \)
53 \( 1 + (-10.1 - 5.85i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.17 + 8.96i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.75 - 6.50i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.00 + 1.15i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.37 + 2.38i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 4.17T + 73T^{2} \)
79 \( 1 + (-13.8 + 8.02i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.57 - 3.79i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.40 + 1.38i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 3.70iT - 97T^{2} \)
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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

−11.38707701912386542128876344415, −10.72132058205843374591549199384, −9.204691677608356398158724185792, −8.706202703916002446447854376547, −7.911561870457990868318902666266, −6.53883659157406213352341778832, −5.17424770449137459232020341007, −3.94315275637081809090197679650, −2.38997653918307609200328063759, −0.35442040400165695902999697219, 1.97653475703771875816134888234, 3.92749557819741820079083849492, 5.48462861497829044418240152805, 6.72319324537084471817397618947, 7.10603595800295529332434858774, 8.292359007113204141662977783237, 9.255252371935305667217763298926, 10.44430963298397615603145384785, 10.88898669659775931858399105151, 11.99352188418744309507191814941

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