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Properties

Label 2-296-296.19-c1-0-31
Degree $2$
Conductor $296$
Sign $-0.783 + 0.621i$
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  + (−0.520 + 1.31i)2-s + (−1.99 − 2.37i)3-s + (−1.45 − 1.36i)4-s + (1.81 − 0.844i)5-s + (4.15 − 1.38i)6-s + (0.0212 − 0.0584i)7-s + (2.55 − 1.20i)8-s + (−1.14 + 6.50i)9-s + (0.167 + 2.81i)10-s + (−2.76 − 1.59i)11-s + (−0.346 + 6.18i)12-s + (−3.32 − 2.32i)13-s + (0.0657 + 0.0584i)14-s + (−5.61 − 2.61i)15-s + (0.250 + 3.99i)16-s + (0.380 − 0.266i)17-s + ⋯
L(s)  = 1  + (−0.368 + 0.929i)2-s + (−1.15 − 1.37i)3-s + (−0.728 − 0.684i)4-s + (0.809 − 0.377i)5-s + (1.69 − 0.564i)6-s + (0.00804 − 0.0220i)7-s + (0.904 − 0.425i)8-s + (−0.382 + 2.16i)9-s + (0.0529 + 0.891i)10-s + (−0.832 − 0.480i)11-s + (−0.0999 + 1.78i)12-s + (−0.922 − 0.646i)13-s + (0.0175 + 0.0156i)14-s + (−1.44 − 0.675i)15-s + (0.0627 + 0.998i)16-s + (0.0923 − 0.0646i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.783 + 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(296\)    =    \(2^{3} \cdot 37\)
Sign: $-0.783 + 0.621i$
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: \((2,\ 296,\ (\ :1/2),\ -0.783 + 0.621i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.120056 - 0.344200i\)
\(L(\frac12)\) \(\approx\) \(0.120056 - 0.344200i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.520 - 1.31i)T \)
37 \( 1 + (-5.84 - 1.66i)T \)
good3 \( 1 + (1.99 + 2.37i)T + (-0.520 + 2.95i)T^{2} \)
5 \( 1 + (-1.81 + 0.844i)T + (3.21 - 3.83i)T^{2} \)
7 \( 1 + (-0.0212 + 0.0584i)T + (-5.36 - 4.49i)T^{2} \)
11 \( 1 + (2.76 + 1.59i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.32 + 2.32i)T + (4.44 + 12.2i)T^{2} \)
17 \( 1 + (-0.380 + 0.266i)T + (5.81 - 15.9i)T^{2} \)
19 \( 1 + (2.95 + 0.258i)T + (18.7 + 3.29i)T^{2} \)
23 \( 1 + (1.78 + 0.477i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (2.62 - 0.702i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (6.47 + 6.47i)T + 31iT^{2} \)
41 \( 1 + (8.93 - 1.57i)T + (38.5 - 14.0i)T^{2} \)
43 \( 1 + (-7.66 - 7.66i)T + 43iT^{2} \)
47 \( 1 + (-3.88 + 2.24i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.44 + 12.2i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (10.0 + 4.69i)T + (37.9 + 45.1i)T^{2} \)
61 \( 1 + (-7.59 + 10.8i)T + (-20.8 - 57.3i)T^{2} \)
67 \( 1 + (1.84 - 5.07i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (-3.29 - 3.93i)T + (-12.3 + 69.9i)T^{2} \)
73 \( 1 + 12.1iT - 73T^{2} \)
79 \( 1 + (-5.48 + 2.55i)T + (50.7 - 60.5i)T^{2} \)
83 \( 1 + (0.0367 - 0.208i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (-4.07 + 8.74i)T + (-57.2 - 68.1i)T^{2} \)
97 \( 1 + (0.678 - 2.53i)T + (-84.0 - 48.5i)T^{2} \)
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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

−11.35496022067860920194846765330, −10.45730486160297790547491243090, −9.451727683068785513372364572263, −8.048083357134723717576641176162, −7.50597478686248736877886008487, −6.34419411025583289389922194741, −5.67824655857317753969936841947, −4.97557211813862004507963920470, −1.98055649221455387126155681893, −0.33353149571140374478749806585, 2.33295020771456267058695574975, 3.94071047033289816794258627455, 4.90711103482798481557915124067, 5.81740879549964214159798857472, 7.27945041224707412553467241721, 8.971163680626607786163875779498, 9.697730014487117596336771124385, 10.42800358637357266174136960894, 10.80216658139637064708001412128, 11.94442480914679501549742122719

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