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Properties

Label 2-290-145.102-c1-0-3
Degree $2$
Conductor $290$
Sign $-0.342 - 0.939i$
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  + (0.781 + 0.623i)2-s + (0.0965 − 0.423i)3-s + (0.222 + 0.974i)4-s + (−1.30 + 1.81i)5-s + (0.339 − 0.270i)6-s + (−4.28 + 2.69i)7-s + (−0.433 + 0.900i)8-s + (2.53 + 1.21i)9-s + (−2.15 + 0.605i)10-s + (2.71 + 0.949i)11-s + 0.434·12-s + (−3.67 − 1.28i)13-s + (−5.03 − 0.567i)14-s + (0.642 + 0.727i)15-s + (−0.900 + 0.433i)16-s − 2.26i·17-s + ⋯
L(s)  = 1  + (0.552 + 0.440i)2-s + (0.0557 − 0.244i)3-s + (0.111 + 0.487i)4-s + (−0.583 + 0.811i)5-s + (0.138 − 0.110i)6-s + (−1.62 + 1.01i)7-s + (−0.153 + 0.318i)8-s + (0.844 + 0.406i)9-s + (−0.680 + 0.191i)10-s + (0.818 + 0.286i)11-s + 0.125·12-s + (−1.01 − 0.356i)13-s + (−1.34 − 0.151i)14-s + (0.165 + 0.187i)15-s + (−0.225 + 0.108i)16-s − 0.549i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(290\)    =    \(2 \cdot 5 \cdot 29\)
Sign: $-0.342 - 0.939i$
Analytic conductor: \(2.31566\)
Root analytic conductor: \(1.52172\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{290} (247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 290,\ (\ :1/2),\ -0.342 - 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.778775 + 1.11260i\)
\(L(\frac12)\) \(\approx\) \(0.778775 + 1.11260i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.781 - 0.623i)T \)
5 \( 1 + (1.30 - 1.81i)T \)
29 \( 1 + (3.76 + 3.85i)T \)
good3 \( 1 + (-0.0965 + 0.423i)T + (-2.70 - 1.30i)T^{2} \)
7 \( 1 + (4.28 - 2.69i)T + (3.03 - 6.30i)T^{2} \)
11 \( 1 + (-2.71 - 0.949i)T + (8.60 + 6.85i)T^{2} \)
13 \( 1 + (3.67 + 1.28i)T + (10.1 + 8.10i)T^{2} \)
17 \( 1 + 2.26iT - 17T^{2} \)
19 \( 1 + (-5.87 - 3.69i)T + (8.24 + 17.1i)T^{2} \)
23 \( 1 + (-6.38 - 0.719i)T + (22.4 + 5.11i)T^{2} \)
31 \( 1 + (-1.67 + 0.188i)T + (30.2 - 6.89i)T^{2} \)
37 \( 1 + (-3.80 - 1.83i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + (8.52 - 8.52i)T - 41iT^{2} \)
43 \( 1 + (-0.177 - 0.222i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (-0.206 + 0.0994i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-0.291 - 2.58i)T + (-51.6 + 11.7i)T^{2} \)
59 \( 1 + 5.72iT - 59T^{2} \)
61 \( 1 + (2.16 - 1.36i)T + (26.4 - 54.9i)T^{2} \)
67 \( 1 + (-10.4 + 3.65i)T + (52.3 - 41.7i)T^{2} \)
71 \( 1 + (-3.51 - 7.28i)T + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (-5.64 + 4.50i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + (-9.84 + 3.44i)T + (61.7 - 49.2i)T^{2} \)
83 \( 1 + (4.56 + 2.86i)T + (36.0 + 74.7i)T^{2} \)
89 \( 1 + (-1.00 - 8.90i)T + (-86.7 + 19.8i)T^{2} \)
97 \( 1 + (-0.384 - 1.68i)T + (-87.3 + 42.0i)T^{2} \)
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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

−12.21285531922401494856205680854, −11.54227002463947290605694311730, −9.949952249582241873851424460889, −9.458454273756409465211338531830, −7.86545024970020967231733838737, −7.03459284492838410190520706973, −6.38447542263481508636681573702, −5.09110928258060715961290009707, −3.58603390673275561899781004190, −2.68544242547157735016137004630, 0.896769406915710949296691871879, 3.33814464946383101151963461415, 4.01445018922807456688009802419, 5.10710558115815217668295886304, 6.74366719283921075519746393868, 7.24549840213072682592579949347, 9.178640693468284827090080517564, 9.540022862352634722982863936264, 10.57403467118498371250291532943, 11.71565331355442745677441028697

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