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Properties

Label 2-290-145.109-c1-0-15
Degree $2$
Conductor $290$
Sign $-0.970 - 0.239i$
Analytic cond. $2.31566$
Root an. cond. $1.52172$
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  + (−0.222 − 0.974i)2-s + (−0.0258 − 0.0124i)3-s + (−0.900 + 0.433i)4-s + (−0.925 − 2.03i)5-s + (−0.00637 + 0.0279i)6-s + (−1.83 + 3.80i)7-s + (0.623 + 0.781i)8-s + (−1.86 − 2.34i)9-s + (−1.77 + 1.35i)10-s + (−3.67 − 2.92i)11-s + 0.0286·12-s + (−3.94 − 3.14i)13-s + (4.11 + 0.939i)14-s + (−0.00142 + 0.0640i)15-s + (0.623 − 0.781i)16-s + 4.51·17-s + ⋯
L(s)  = 1  + (−0.157 − 0.689i)2-s + (−0.0148 − 0.00717i)3-s + (−0.450 + 0.216i)4-s + (−0.413 − 0.910i)5-s + (−0.00260 + 0.0113i)6-s + (−0.692 + 1.43i)7-s + (0.220 + 0.276i)8-s + (−0.623 − 0.781i)9-s + (−0.562 + 0.428i)10-s + (−1.10 − 0.882i)11-s + 0.00826·12-s + (−1.09 − 0.872i)13-s + (1.09 + 0.250i)14-s + (−0.000367 + 0.0165i)15-s + (0.155 − 0.195i)16-s + 1.09·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.239i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 - 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(290\)    =    \(2 \cdot 5 \cdot 29\)
Sign: $-0.970 - 0.239i$
Analytic conductor: \(2.31566\)
Root analytic conductor: \(1.52172\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{290} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 290,\ (\ :1/2),\ -0.970 - 0.239i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0420827 + 0.346775i\)
\(L(\frac12)\) \(\approx\) \(0.0420827 + 0.346775i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.222 + 0.974i)T \)
5 \( 1 + (0.925 + 2.03i)T \)
29 \( 1 + (-2.93 + 4.51i)T \)
good3 \( 1 + (0.0258 + 0.0124i)T + (1.87 + 2.34i)T^{2} \)
7 \( 1 + (1.83 - 3.80i)T + (-4.36 - 5.47i)T^{2} \)
11 \( 1 + (3.67 + 2.92i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (3.94 + 3.14i)T + (2.89 + 12.6i)T^{2} \)
17 \( 1 - 4.51T + 17T^{2} \)
19 \( 1 + (-2.57 - 5.35i)T + (-11.8 + 14.8i)T^{2} \)
23 \( 1 + (7.99 + 1.82i)T + (20.7 + 9.97i)T^{2} \)
31 \( 1 + (-0.562 + 0.128i)T + (27.9 - 13.4i)T^{2} \)
37 \( 1 + (1.08 + 1.36i)T + (-8.23 + 36.0i)T^{2} \)
41 \( 1 - 1.09iT - 41T^{2} \)
43 \( 1 + (-1.21 + 5.34i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (0.174 - 0.218i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-0.286 + 0.0654i)T + (47.7 - 22.9i)T^{2} \)
59 \( 1 - 11.2T + 59T^{2} \)
61 \( 1 + (-3.48 + 7.24i)T + (-38.0 - 47.6i)T^{2} \)
67 \( 1 + (-0.131 + 0.105i)T + (14.9 - 65.3i)T^{2} \)
71 \( 1 + (-4.47 + 5.61i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-0.285 + 1.25i)T + (-65.7 - 31.6i)T^{2} \)
79 \( 1 + (0.931 - 0.742i)T + (17.5 - 77.0i)T^{2} \)
83 \( 1 + (3.34 + 6.95i)T + (-51.7 + 64.8i)T^{2} \)
89 \( 1 + (11.8 - 2.69i)T + (80.1 - 38.6i)T^{2} \)
97 \( 1 + (2.16 - 1.04i)T + (60.4 - 75.8i)T^{2} \)
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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

−11.66810898980723072286066253940, −10.12132799586066239743005691950, −9.568952028152546232913930107746, −8.360877199601974757405488882201, −8.004241074777995656503817150337, −5.78613280801223249063182354714, −5.42421731062419786186842332565, −3.57892540404695965513103956816, −2.59365229651044524317708808063, −0.25825701226774622139980524589, 2.72250429051487844075235798362, 4.18382025467208783817079827095, 5.31216964397854557385028147998, 6.85050777891158611431802716590, 7.33887006999454381907139874963, 8.043975683421953704885393952974, 9.855663781312547239346948743406, 10.11643878971675736545999135609, 11.18335909330218779159519194336, 12.32646398752892442683781834225

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