Properties

Label 2-6035-1.1-c1-0-157
Degree $2$
Conductor $6035$
Sign $-1$
Analytic cond. $48.1897$
Root an. cond. $6.94188$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.404·2-s − 1.99·3-s − 1.83·4-s + 5-s − 0.805·6-s − 2.54·7-s − 1.55·8-s + 0.968·9-s + 0.404·10-s − 3.71·11-s + 3.65·12-s − 4.88·13-s − 1.02·14-s − 1.99·15-s + 3.04·16-s + 17-s + 0.391·18-s − 2.52·19-s − 1.83·20-s + 5.06·21-s − 1.50·22-s + 5.94·23-s + 3.08·24-s + 25-s − 1.97·26-s + 4.04·27-s + 4.66·28-s + ⋯
L(s)  = 1  + 0.285·2-s − 1.15·3-s − 0.918·4-s + 0.447·5-s − 0.328·6-s − 0.960·7-s − 0.548·8-s + 0.322·9-s + 0.127·10-s − 1.11·11-s + 1.05·12-s − 1.35·13-s − 0.274·14-s − 0.514·15-s + 0.761·16-s + 0.242·17-s + 0.0922·18-s − 0.578·19-s − 0.410·20-s + 1.10·21-s − 0.319·22-s + 1.23·23-s + 0.630·24-s + 0.200·25-s − 0.387·26-s + 0.778·27-s + 0.882·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6035\)    =    \(5 \cdot 17 \cdot 71\)
Sign: $-1$
Analytic conductor: \(48.1897\)
Root analytic conductor: \(6.94188\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6035,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
17 \( 1 - T \)
71 \( 1 + T \)
good2 \( 1 - 0.404T + 2T^{2} \)
3 \( 1 + 1.99T + 3T^{2} \)
7 \( 1 + 2.54T + 7T^{2} \)
11 \( 1 + 3.71T + 11T^{2} \)
13 \( 1 + 4.88T + 13T^{2} \)
19 \( 1 + 2.52T + 19T^{2} \)
23 \( 1 - 5.94T + 23T^{2} \)
29 \( 1 - 9.14T + 29T^{2} \)
31 \( 1 - 9.91T + 31T^{2} \)
37 \( 1 + 1.61T + 37T^{2} \)
41 \( 1 + 3.99T + 41T^{2} \)
43 \( 1 - 5.28T + 43T^{2} \)
47 \( 1 + 8.72T + 47T^{2} \)
53 \( 1 - 8.60T + 53T^{2} \)
59 \( 1 + 0.753T + 59T^{2} \)
61 \( 1 + 13.1T + 61T^{2} \)
67 \( 1 + 2.42T + 67T^{2} \)
73 \( 1 - 12.7T + 73T^{2} \)
79 \( 1 - 9.28T + 79T^{2} \)
83 \( 1 - 7.27T + 83T^{2} \)
89 \( 1 - 1.63T + 89T^{2} \)
97 \( 1 - 1.68T + 97T^{2} \)
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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

−7.69994962471070853745212324050, −6.54825546504459574868135285019, −6.39845632459549251295694715936, −5.29728290330925721515045135003, −5.05181480351169707635531130435, −4.42781969728380576345798946393, −3.07337376413837237228200490643, −2.64666105972537806919426349420, −0.851526624198747068817892316293, 0, 0.851526624198747068817892316293, 2.64666105972537806919426349420, 3.07337376413837237228200490643, 4.42781969728380576345798946393, 5.05181480351169707635531130435, 5.29728290330925721515045135003, 6.39845632459549251295694715936, 6.54825546504459574868135285019, 7.69994962471070853745212324050

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