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Properties

Label 2-294-49.9-c1-0-0
Degree $2$
Conductor $294$
Sign $-0.528 - 0.849i$
Analytic cond. $2.34760$
Root an. cond. $1.53218$
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  + (−0.733 − 0.680i)2-s + (−0.988 − 0.149i)3-s + (0.0747 + 0.997i)4-s + (−1.49 − 3.81i)5-s + (0.623 + 0.781i)6-s + (−0.683 + 2.55i)7-s + (0.623 − 0.781i)8-s + (0.955 + 0.294i)9-s + (−1.49 + 3.81i)10-s + (−4.24 + 1.30i)11-s + (0.0747 − 0.997i)12-s + (0.268 − 1.17i)13-s + (2.23 − 1.40i)14-s + (0.912 + 3.99i)15-s + (−0.988 + 0.149i)16-s + (2.98 + 2.03i)17-s + ⋯
L(s)  = 1  + (−0.518 − 0.480i)2-s + (−0.570 − 0.0860i)3-s + (0.0373 + 0.498i)4-s + (−0.670 − 1.70i)5-s + (0.254 + 0.319i)6-s + (−0.258 + 0.966i)7-s + (0.220 − 0.276i)8-s + (0.318 + 0.0982i)9-s + (−0.473 + 1.20i)10-s + (−1.27 + 0.394i)11-s + (0.0215 − 0.287i)12-s + (0.0745 − 0.326i)13-s + (0.598 − 0.376i)14-s + (0.235 + 1.03i)15-s + (−0.247 + 0.0372i)16-s + (0.724 + 0.493i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.528 - 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.528 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $-0.528 - 0.849i$
Analytic conductor: \(2.34760\)
Root analytic conductor: \(1.53218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :1/2),\ -0.528 - 0.849i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00725194 + 0.0130525i\)
\(L(\frac12)\) \(\approx\) \(0.00725194 + 0.0130525i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.733 + 0.680i)T \)
3 \( 1 + (0.988 + 0.149i)T \)
7 \( 1 + (0.683 - 2.55i)T \)
good5 \( 1 + (1.49 + 3.81i)T + (-3.66 + 3.40i)T^{2} \)
11 \( 1 + (4.24 - 1.30i)T + (9.08 - 6.19i)T^{2} \)
13 \( 1 + (-0.268 + 1.17i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 + (-2.98 - 2.03i)T + (6.21 + 15.8i)T^{2} \)
19 \( 1 + (2.10 - 3.63i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.13 - 4.18i)T + (8.40 - 21.4i)T^{2} \)
29 \( 1 + (0.564 - 0.271i)T + (18.0 - 22.6i)T^{2} \)
31 \( 1 + (3.64 + 6.30i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.162 + 2.16i)T + (-36.5 - 5.51i)T^{2} \)
41 \( 1 + (3.15 - 3.95i)T + (-9.12 - 39.9i)T^{2} \)
43 \( 1 + (-1.92 - 2.40i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (0.154 + 0.143i)T + (3.51 + 46.8i)T^{2} \)
53 \( 1 + (0.299 + 3.99i)T + (-52.4 + 7.89i)T^{2} \)
59 \( 1 + (-5.59 + 14.2i)T + (-43.2 - 40.1i)T^{2} \)
61 \( 1 + (-0.253 + 3.38i)T + (-60.3 - 9.09i)T^{2} \)
67 \( 1 + (0.294 + 0.509i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (12.1 + 5.84i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (-5.40 + 5.01i)T + (5.45 - 72.7i)T^{2} \)
79 \( 1 + (-1.23 + 2.13i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.48 - 6.50i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (11.9 + 3.69i)T + (73.5 + 50.1i)T^{2} \)
97 \( 1 + 9.91T + 97T^{2} \)
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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

−12.18337270777377789945808940845, −11.35971598282573764859719442063, −10.09638781067317346118662229125, −9.358418750783864537824186756792, −8.086345835221634814570118026360, −7.913586478210630427192449685664, −5.85522141040099567573397005161, −5.10764436252722750080701337266, −3.79813608286293023527858269783, −1.84020377911020300694366098440, 0.01345579586578896387358296776, 2.83122377527293464648498066408, 4.18397811848438210544727615436, 5.73279487313466035325227757724, 6.87784259154546330372732983816, 7.28574305330431060752141375800, 8.306904942407999832132877912939, 9.984794943388904765608565953685, 10.56033979931790367485057847888, 11.02762650611744661865771058575

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