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Properties

Label 2-3017-3017.1392-c0-0-0
Degree $2$
Conductor $3017$
Sign $-0.995 + 0.0923i$
Analytic cond. $1.50567$
Root an. cond. $1.22706$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.230 − 0.543i)2-s + (0.452 − 0.469i)4-s + (−0.650 − 0.759i)7-s + (−0.910 − 0.348i)8-s + (0.520 − 0.853i)9-s + (−1.21 + 1.59i)11-s + (−0.262 + 0.528i)14-s + (−0.00279 − 0.0763i)16-s + (−0.583 − 0.0859i)18-s + (1.14 + 0.290i)22-s + (−0.129 − 1.03i)23-s + (0.593 − 0.804i)25-s + (−0.650 − 0.0380i)28-s + (−1.32 − 0.234i)29-s + (−0.923 + 0.432i)32-s + ⋯
L(s)  = 1  + (−0.230 − 0.543i)2-s + (0.452 − 0.469i)4-s + (−0.650 − 0.759i)7-s + (−0.910 − 0.348i)8-s + (0.520 − 0.853i)9-s + (−1.21 + 1.59i)11-s + (−0.262 + 0.528i)14-s + (−0.00279 − 0.0763i)16-s + (−0.583 − 0.0859i)18-s + (1.14 + 0.290i)22-s + (−0.129 − 1.03i)23-s + (0.593 − 0.804i)25-s + (−0.650 − 0.0380i)28-s + (−1.32 − 0.234i)29-s + (−0.923 + 0.432i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3017\)    =    \(7 \cdot 431\)
Sign: $-0.995 + 0.0923i$
Analytic conductor: \(1.50567\)
Root analytic conductor: \(1.22706\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3017} (1392, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3017,\ (\ :0),\ -0.995 + 0.0923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7234715659\)
\(L(\frac12)\) \(\approx\) \(0.7234715659\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.650 + 0.759i)T \)
431 \( 1 + (-0.639 + 0.768i)T \)
good2 \( 1 + (0.230 + 0.543i)T + (-0.694 + 0.719i)T^{2} \)
3 \( 1 + (-0.520 + 0.853i)T^{2} \)
5 \( 1 + (-0.593 + 0.804i)T^{2} \)
11 \( 1 + (1.21 - 1.59i)T + (-0.267 - 0.963i)T^{2} \)
13 \( 1 + (-0.470 + 0.882i)T^{2} \)
17 \( 1 + (0.999 + 0.0438i)T^{2} \)
19 \( 1 + (0.00730 + 0.999i)T^{2} \)
23 \( 1 + (0.129 + 1.03i)T + (-0.969 + 0.245i)T^{2} \)
29 \( 1 + (1.32 + 0.234i)T + (0.939 + 0.343i)T^{2} \)
31 \( 1 + (-0.364 - 0.931i)T^{2} \)
37 \( 1 + (1.39 + 1.36i)T + (0.0219 + 0.999i)T^{2} \)
41 \( 1 + (-0.763 + 0.645i)T^{2} \)
43 \( 1 + (-1.27 + 1.07i)T + (0.167 - 0.985i)T^{2} \)
47 \( 1 + (0.934 - 0.357i)T^{2} \)
53 \( 1 + (0.980 + 0.531i)T + (0.545 + 0.838i)T^{2} \)
59 \( 1 + (-0.138 + 0.990i)T^{2} \)
61 \( 1 + (0.982 - 0.188i)T^{2} \)
67 \( 1 + (1.00 - 0.528i)T + (0.569 - 0.821i)T^{2} \)
71 \( 1 + (0.168 - 0.353i)T + (-0.628 - 0.777i)T^{2} \)
73 \( 1 + (-0.917 - 0.397i)T^{2} \)
79 \( 1 + (0.313 - 1.45i)T + (-0.911 - 0.411i)T^{2} \)
83 \( 1 + (0.650 + 0.759i)T^{2} \)
89 \( 1 + (-0.281 + 0.959i)T^{2} \)
97 \( 1 + (-0.979 - 0.203i)T^{2} \)
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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

−8.824746138008303018445183530882, −7.48391703168063125090114583596, −7.12712292325860606185716318204, −6.42194354725714398275002436145, −5.52642699508914994229257827242, −4.49959884879756688951302035031, −3.70489452865727491894465950397, −2.63784959809226019237353171344, −1.84619604180037546542721979381, −0.43631417301134157541856693915, 1.87493867741352837064067392618, 3.04409675127279364275804907694, 3.32639463227278610087202344259, 4.91936251413940629957818048738, 5.70735222460455800783536746286, 6.12768691092571967636434711429, 7.25799830154959652749843304428, 7.67579899916812040782120866389, 8.462665827388246848772942429056, 8.995456357819612155881678007310

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