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Properties

Label 2-3017-3017.1035-c0-0-0
Degree $2$
Conductor $3017$
Sign $-0.0612 + 0.998i$
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  + (−0.914 − 1.09i)2-s + (−0.189 + 1.02i)4-s + (−0.899 + 0.437i)7-s + (0.0546 − 0.0306i)8-s + (−0.694 − 0.719i)9-s + (−0.801 + 0.907i)11-s + (1.30 + 0.587i)14-s + (0.891 + 0.340i)16-s + (−0.156 + 1.42i)18-s + (1.73 + 0.0505i)22-s + (1.38 − 0.0202i)23-s + (−0.377 − 0.925i)25-s + (−0.278 − 1.00i)28-s + (1.29 + 1.22i)29-s + (−0.460 − 1.35i)32-s + ⋯
L(s)  = 1  + (−0.914 − 1.09i)2-s + (−0.189 + 1.02i)4-s + (−0.899 + 0.437i)7-s + (0.0546 − 0.0306i)8-s + (−0.694 − 0.719i)9-s + (−0.801 + 0.907i)11-s + (1.30 + 0.587i)14-s + (0.891 + 0.340i)16-s + (−0.156 + 1.42i)18-s + (1.73 + 0.0505i)22-s + (1.38 − 0.0202i)23-s + (−0.377 − 0.925i)25-s + (−0.278 − 1.00i)28-s + (1.29 + 1.22i)29-s + (−0.460 − 1.35i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0612 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0612 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3017\)    =    \(7 \cdot 431\)
Sign: $-0.0612 + 0.998i$
Analytic conductor: \(1.50567\)
Root analytic conductor: \(1.22706\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3017} (1035, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3017,\ (\ :0),\ -0.0612 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5134211980\)
\(L(\frac12)\) \(\approx\) \(0.5134211980\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.899 - 0.437i)T \)
431 \( 1 + (0.791 - 0.611i)T \)
good2 \( 1 + (0.914 + 1.09i)T + (-0.181 + 0.983i)T^{2} \)
3 \( 1 + (0.694 + 0.719i)T^{2} \)
5 \( 1 + (0.377 + 0.925i)T^{2} \)
11 \( 1 + (0.801 - 0.907i)T + (-0.123 - 0.992i)T^{2} \)
13 \( 1 + (-0.879 + 0.476i)T^{2} \)
17 \( 1 + (0.982 + 0.188i)T^{2} \)
19 \( 1 + (-0.849 + 0.527i)T^{2} \)
23 \( 1 + (-1.38 + 0.0202i)T + (0.999 - 0.0292i)T^{2} \)
29 \( 1 + (-1.29 - 1.22i)T + (0.0511 + 0.998i)T^{2} \)
31 \( 1 + (0.999 - 0.0438i)T^{2} \)
37 \( 1 + (-1.32 + 1.46i)T + (-0.0948 - 0.995i)T^{2} \)
41 \( 1 + (0.994 + 0.102i)T^{2} \)
43 \( 1 + (1.80 + 0.184i)T + (0.979 + 0.203i)T^{2} \)
47 \( 1 + (0.872 + 0.489i)T^{2} \)
53 \( 1 + (-1.08 - 0.0637i)T + (0.993 + 0.116i)T^{2} \)
59 \( 1 + (0.431 + 0.902i)T^{2} \)
61 \( 1 + (0.295 + 0.955i)T^{2} \)
67 \( 1 + (0.504 + 0.883i)T + (-0.508 + 0.861i)T^{2} \)
71 \( 1 + (-1.16 + 0.956i)T + (0.195 - 0.980i)T^{2} \)
73 \( 1 + (-0.948 + 0.315i)T^{2} \)
79 \( 1 + (0.961 + 0.400i)T + (0.704 + 0.709i)T^{2} \)
83 \( 1 + (0.899 - 0.437i)T^{2} \)
89 \( 1 + (0.944 - 0.329i)T^{2} \)
97 \( 1 + (0.987 - 0.160i)T^{2} \)
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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

−8.876307610074208993450688907752, −8.397469344768285487505588905781, −7.34685522084060318990359450457, −6.50728713976479240600089365263, −5.71691696131487554306856836753, −4.76518350107170724703267970886, −3.45098015183254664168420373953, −2.86495686465232042875164897666, −2.11101786538213253298273399941, −0.62467810737701412641534346631, 0.817392288270833033344297694042, 2.74047711749085246818085148353, 3.32361610563654291815814903315, 4.75510473967197283897261852640, 5.62654901722832695097689338787, 6.20857013280758118347210026702, 6.97243290728933102125442013134, 7.65679495244139043330966758225, 8.400138574197563317451365110828, 8.722385914858642670076988555137

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