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Properties

Label 2-296-296.101-c1-0-11
Degree $2$
Conductor $296$
Sign $-0.591 - 0.806i$
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  + (0.183 + 1.40i)2-s + (0.576 − 0.332i)3-s + (−1.93 + 0.514i)4-s + (1.60 + 2.78i)5-s + (0.572 + 0.746i)6-s + (−0.874 − 1.51i)7-s + (−1.07 − 2.61i)8-s + (−1.27 + 2.21i)9-s + (−3.61 + 2.76i)10-s + 3.96i·11-s + (−0.942 + 0.939i)12-s + (−0.268 − 0.465i)13-s + (1.96 − 1.50i)14-s + (1.85 + 1.07i)15-s + (3.46 − 1.99i)16-s + (6.04 + 3.48i)17-s + ⋯
L(s)  = 1  + (0.129 + 0.991i)2-s + (0.332 − 0.192i)3-s + (−0.966 + 0.257i)4-s + (0.719 + 1.24i)5-s + (0.233 + 0.304i)6-s + (−0.330 − 0.572i)7-s + (−0.380 − 0.924i)8-s + (−0.426 + 0.738i)9-s + (−1.14 + 0.875i)10-s + 1.19i·11-s + (−0.271 + 0.271i)12-s + (−0.0744 − 0.128i)13-s + (0.524 − 0.401i)14-s + (0.478 + 0.276i)15-s + (0.867 − 0.497i)16-s + (1.46 + 0.845i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.591 - 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.591 - 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(296\)    =    \(2^{3} \cdot 37\)
Sign: $-0.591 - 0.806i$
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: \((2,\ 296,\ (\ :1/2),\ -0.591 - 0.806i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.635694 + 1.25481i\)
\(L(\frac12)\) \(\approx\) \(0.635694 + 1.25481i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.183 - 1.40i)T \)
37 \( 1 + (0.0791 + 6.08i)T \)
good3 \( 1 + (-0.576 + 0.332i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.60 - 2.78i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.874 + 1.51i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 3.96iT - 11T^{2} \)
13 \( 1 + (0.268 + 0.465i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-6.04 - 3.48i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.21 + 3.84i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 2.14iT - 23T^{2} \)
29 \( 1 + 3.76T + 29T^{2} \)
31 \( 1 + 1.95iT - 31T^{2} \)
41 \( 1 + (-1.70 - 2.95i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 7.07T + 43T^{2} \)
47 \( 1 - 8.02T + 47T^{2} \)
53 \( 1 + (0.341 + 0.197i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.26 + 7.38i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.607 + 1.05i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.25 + 4.18i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.54 + 4.40i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 2.30T + 73T^{2} \)
79 \( 1 + (2.04 - 1.18i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.40 + 0.812i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-13.0 - 7.51i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 16.7iT - 97T^{2} \)
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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

−12.50007146245571954078516822921, −10.81156952810219857279947696702, −10.14338826671353581881991487382, −9.249667046755132491821715055843, −7.86482355552125249130844623138, −7.25031297758451518635674294996, −6.36212497362721951135410349299, −5.31920116024174605585143279781, −3.84883568667817691577728821794, −2.44354022649442880029127355255, 1.06467923377507276004379393623, 2.78483988902177535334455268988, 3.88976885059781440428835513831, 5.44989370932715167663822058799, 5.85038044515538124174414475983, 8.127182070211530543006151305853, 9.030671075082153214774328235478, 9.374736558872012846175254900924, 10.37376475658266962206017782044, 11.74118539658440950414977549602

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