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Properties

Label 2-296-296.101-c1-0-14
Degree $2$
Conductor $296$
Sign $-0.278 + 0.960i$
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  + (−0.262 − 1.38i)2-s + (−2.44 + 1.41i)3-s + (−1.86 + 0.729i)4-s + (−0.380 − 0.658i)5-s + (2.60 + 3.02i)6-s + (1.51 + 2.62i)7-s + (1.50 + 2.39i)8-s + (2.48 − 4.29i)9-s + (−0.815 + 0.701i)10-s − 4.75i·11-s + (3.52 − 4.40i)12-s + (−2.66 − 4.61i)13-s + (3.25 − 2.79i)14-s + (1.85 + 1.07i)15-s + (2.93 − 2.71i)16-s + (−3.24 − 1.87i)17-s + ⋯
L(s)  = 1  + (−0.185 − 0.982i)2-s + (−1.41 + 0.814i)3-s + (−0.931 + 0.364i)4-s + (−0.170 − 0.294i)5-s + (1.06 + 1.23i)6-s + (0.573 + 0.992i)7-s + (0.531 + 0.847i)8-s + (0.826 − 1.43i)9-s + (−0.257 + 0.221i)10-s − 1.43i·11-s + (1.01 − 1.27i)12-s + (−0.739 − 1.28i)13-s + (0.869 − 0.747i)14-s + (0.479 + 0.276i)15-s + (0.733 − 0.679i)16-s + (−0.787 − 0.454i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.278 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.278 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(296\)    =    \(2^{3} \cdot 37\)
Sign: $-0.278 + 0.960i$
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: \((2,\ 296,\ (\ :1/2),\ -0.278 + 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.319038 - 0.424615i\)
\(L(\frac12)\) \(\approx\) \(0.319038 - 0.424615i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.262 + 1.38i)T \)
37 \( 1 + (-5.34 + 2.91i)T \)
good3 \( 1 + (2.44 - 1.41i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.380 + 0.658i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.51 - 2.62i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 4.75iT - 11T^{2} \)
13 \( 1 + (2.66 + 4.61i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.24 + 1.87i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.18 - 7.24i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 6.95iT - 23T^{2} \)
29 \( 1 - 5.45T + 29T^{2} \)
31 \( 1 + 2.47iT - 31T^{2} \)
41 \( 1 + (1.20 + 2.08i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 3.43T + 43T^{2} \)
47 \( 1 + 0.977T + 47T^{2} \)
53 \( 1 + (-3.03 - 1.75i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.51 + 2.62i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.770 + 1.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (13.2 - 7.62i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.41 + 2.44i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 7.88T + 73T^{2} \)
79 \( 1 + (-0.687 + 0.396i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.50 + 1.44i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.62 - 2.09i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.59iT - 97T^{2} \)
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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

−11.55076174733442648005632441514, −10.59839683012441204096133330846, −10.06629923593344649735793641072, −8.808827671858120270439546161917, −8.064992038225218371004086792828, −5.96300790543034446828084188869, −5.28668152890899414911342362321, −4.38945347219483317218023069162, −2.87136091274871013899434230601, −0.56093804485960841270991800833, 1.40122956716231647079220872803, 4.53288273772893818911606069326, 4.98502021602100293408529447562, 6.51690298363056660589292329906, 7.20562760377874408525757348885, 7.44065492779277995091980327356, 9.196149007190716080198677088688, 10.20694644679662909972043740533, 11.25066003247886927330810355450, 11.90119710125067511013856901677

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