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Properties

Label 2-3017-3017.139-c0-0-0
Degree $2$
Conductor $3017$
Sign $0.691 - 0.722i$
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  + (0.990 − 0.655i)2-s + (0.160 − 0.377i)4-s + (0.417 + 0.908i)7-s + (0.127 + 0.689i)8-s + (−0.872 − 0.489i)9-s + (−0.888 + 1.79i)11-s + (1.00 + 0.625i)14-s + (0.861 + 0.894i)16-s + (−1.18 + 0.0866i)18-s + (0.294 + 2.35i)22-s + (−1.15 − 1.30i)23-s + (0.892 + 0.450i)25-s + (0.409 − 0.0119i)28-s + (−0.805 + 0.0708i)29-s + (0.755 + 0.168i)32-s + ⋯
L(s)  = 1  + (0.990 − 0.655i)2-s + (0.160 − 0.377i)4-s + (0.417 + 0.908i)7-s + (0.127 + 0.689i)8-s + (−0.872 − 0.489i)9-s + (−0.888 + 1.79i)11-s + (1.00 + 0.625i)14-s + (0.861 + 0.894i)16-s + (−1.18 + 0.0866i)18-s + (0.294 + 2.35i)22-s + (−1.15 − 1.30i)23-s + (0.892 + 0.450i)25-s + (0.409 − 0.0119i)28-s + (−0.805 + 0.0708i)29-s + (0.755 + 0.168i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3017\)    =    \(7 \cdot 431\)
Sign: $0.691 - 0.722i$
Analytic conductor: \(1.50567\)
Root analytic conductor: \(1.22706\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3017} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3017,\ (\ :0),\ 0.691 - 0.722i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.744208921\)
\(L(\frac12)\) \(\approx\) \(1.744208921\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.417 - 0.908i)T \)
431 \( 1 + (-0.905 - 0.424i)T \)
good2 \( 1 + (-0.990 + 0.655i)T + (0.391 - 0.920i)T^{2} \)
3 \( 1 + (0.872 + 0.489i)T^{2} \)
5 \( 1 + (-0.892 - 0.450i)T^{2} \)
11 \( 1 + (0.888 - 1.79i)T + (-0.605 - 0.795i)T^{2} \)
13 \( 1 + (0.857 + 0.514i)T^{2} \)
17 \( 1 + (-0.0219 - 0.999i)T^{2} \)
19 \( 1 + (-0.704 - 0.709i)T^{2} \)
23 \( 1 + (1.15 + 1.30i)T + (-0.123 + 0.992i)T^{2} \)
29 \( 1 + (0.805 - 0.0708i)T + (0.984 - 0.174i)T^{2} \)
31 \( 1 + (0.825 - 0.563i)T^{2} \)
37 \( 1 + (-0.751 - 1.84i)T + (-0.714 + 0.699i)T^{2} \)
41 \( 1 + (-0.939 - 0.343i)T^{2} \)
43 \( 1 + (-1.79 - 0.657i)T + (0.763 + 0.645i)T^{2} \)
47 \( 1 + (0.181 - 0.983i)T^{2} \)
53 \( 1 + (0.911 - 0.231i)T + (0.879 - 0.476i)T^{2} \)
59 \( 1 + (0.754 + 0.656i)T^{2} \)
61 \( 1 + (0.0948 - 0.995i)T^{2} \)
67 \( 1 + (-0.423 + 1.72i)T + (-0.885 - 0.463i)T^{2} \)
71 \( 1 + (-0.824 + 1.30i)T + (-0.431 - 0.902i)T^{2} \)
73 \( 1 + (-0.979 + 0.203i)T^{2} \)
79 \( 1 + (-1.17 + 1.45i)T + (-0.210 - 0.977i)T^{2} \)
83 \( 1 + (-0.417 - 0.908i)T^{2} \)
89 \( 1 + (-0.800 - 0.599i)T^{2} \)
97 \( 1 + (0.994 - 0.102i)T^{2} \)
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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

−9.059976223335014174806724648770, −8.154370123586112602773211918177, −7.70804866287138448371531029541, −6.43639075579976191827881263012, −5.76013424531657079422115278640, −4.78663762750643482745281566496, −4.60028352993312992379395107881, −3.29030737728894902921826660824, −2.52899640210389140841389755377, −1.92261352708207932315081459355, 0.75634857041225980040477136312, 2.47397443291391048326317993581, 3.54836519131107109412114808846, 4.09573651717452390793124106085, 5.27538064621575821712561101108, 5.59039168777965941851159491433, 6.24662260750210747040596041580, 7.36496646536965199537372851046, 7.82778342881674214434853303801, 8.542071686782682919880738056687

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