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Properties

Label 2-296-296.101-c1-0-16
Degree $2$
Conductor $296$
Sign $0.996 + 0.0816i$
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  + (1.06 − 0.927i)2-s + (−0.280 + 0.161i)3-s + (0.279 − 1.98i)4-s + (1.57 + 2.73i)5-s + (−0.149 + 0.432i)6-s + (2.58 + 4.47i)7-s + (−1.53 − 2.37i)8-s + (−1.44 + 2.50i)9-s + (4.21 + 1.45i)10-s − 2.54i·11-s + (0.242 + 0.600i)12-s + (−1.47 − 2.55i)13-s + (6.91 + 2.38i)14-s + (−0.884 − 0.510i)15-s + (−3.84 − 1.10i)16-s + (1.46 + 0.844i)17-s + ⋯
L(s)  = 1  + (0.754 − 0.655i)2-s + (−0.161 + 0.0934i)3-s + (0.139 − 0.990i)4-s + (0.705 + 1.22i)5-s + (−0.0609 + 0.176i)6-s + (0.977 + 1.69i)7-s + (−0.543 − 0.839i)8-s + (−0.482 + 0.835i)9-s + (1.33 + 0.459i)10-s − 0.768i·11-s + (0.0699 + 0.173i)12-s + (−0.409 − 0.708i)13-s + (1.84 + 0.637i)14-s + (−0.228 − 0.131i)15-s + (−0.960 − 0.276i)16-s + (0.354 + 0.204i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(296\)    =    \(2^{3} \cdot 37\)
Sign: $0.996 + 0.0816i$
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: \((2,\ 296,\ (\ :1/2),\ 0.996 + 0.0816i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.02928 - 0.0830180i\)
\(L(\frac12)\) \(\approx\) \(2.02928 - 0.0830180i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.06 + 0.927i)T \)
37 \( 1 + (-3.84 - 4.71i)T \)
good3 \( 1 + (0.280 - 0.161i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.57 - 2.73i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-2.58 - 4.47i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 2.54iT - 11T^{2} \)
13 \( 1 + (1.47 + 2.55i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.46 - 0.844i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.12 + 5.40i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 4.40iT - 23T^{2} \)
29 \( 1 - 0.503T + 29T^{2} \)
31 \( 1 + 9.67iT - 31T^{2} \)
41 \( 1 + (-1.45 - 2.52i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 4.61T + 43T^{2} \)
47 \( 1 - 1.20T + 47T^{2} \)
53 \( 1 + (-0.876 - 0.505i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.51 - 4.35i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.58 + 7.95i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.154 - 0.0890i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.55 - 7.89i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 0.194T + 73T^{2} \)
79 \( 1 + (-5.31 + 3.06i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.26 + 2.46i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (9.83 + 5.67i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 1.65iT - 97T^{2} \)
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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

−11.46036664709629794901222566203, −11.09853642973964549912947183314, −10.27132940054665673778313902275, −9.095249760728660872193126504161, −7.989050137053876669901530604230, −6.27114569432601625991173674411, −5.71120031079963666489603425779, −4.77453233824156399926107159827, −2.68510422376379358395786656303, −2.41416347237165846831550164009, 1.54893036799606211730991398807, 3.89204934913179481368236951032, 4.69186711312324502042085516056, 5.62657215793961231024434372219, 6.86935647838544780025364864091, 7.71663872153858695567685090403, 8.757817842494589074780228082436, 9.782663912567359080381604545388, 11.03896148305943604175566275544, 12.18982888722095410563315683736

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