Properties

Label 2-4024-1.1-c1-0-57
Degree $2$
Conductor $4024$
Sign $-1$
Analytic cond. $32.1318$
Root an. cond. $5.66849$
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  − 2.34·3-s − 1.32·5-s + 0.439·7-s + 2.48·9-s + 1.15·11-s − 2.14·13-s + 3.10·15-s + 2.70·17-s − 1.78·19-s − 1.03·21-s − 2.60·23-s − 3.24·25-s + 1.19·27-s + 5.83·29-s − 1.27·31-s − 2.69·33-s − 0.582·35-s − 0.680·37-s + 5.02·39-s + 2.61·41-s + 2.38·43-s − 3.29·45-s − 5.77·47-s − 6.80·49-s − 6.33·51-s + 9.90·53-s − 1.52·55-s + ⋯
L(s)  = 1  − 1.35·3-s − 0.592·5-s + 0.166·7-s + 0.829·9-s + 0.347·11-s − 0.594·13-s + 0.801·15-s + 0.655·17-s − 0.409·19-s − 0.224·21-s − 0.543·23-s − 0.649·25-s + 0.230·27-s + 1.08·29-s − 0.228·31-s − 0.469·33-s − 0.0984·35-s − 0.111·37-s + 0.803·39-s + 0.408·41-s + 0.363·43-s − 0.491·45-s − 0.841·47-s − 0.972·49-s − 0.886·51-s + 1.36·53-s − 0.205·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $-1$
Analytic conductor: \(32.1318\)
Root analytic conductor: \(5.66849\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4024,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 \)
503 \( 1 + T \)
good3 \( 1 + 2.34T + 3T^{2} \)
5 \( 1 + 1.32T + 5T^{2} \)
7 \( 1 - 0.439T + 7T^{2} \)
11 \( 1 - 1.15T + 11T^{2} \)
13 \( 1 + 2.14T + 13T^{2} \)
17 \( 1 - 2.70T + 17T^{2} \)
19 \( 1 + 1.78T + 19T^{2} \)
23 \( 1 + 2.60T + 23T^{2} \)
29 \( 1 - 5.83T + 29T^{2} \)
31 \( 1 + 1.27T + 31T^{2} \)
37 \( 1 + 0.680T + 37T^{2} \)
41 \( 1 - 2.61T + 41T^{2} \)
43 \( 1 - 2.38T + 43T^{2} \)
47 \( 1 + 5.77T + 47T^{2} \)
53 \( 1 - 9.90T + 53T^{2} \)
59 \( 1 - 5.85T + 59T^{2} \)
61 \( 1 - 13.8T + 61T^{2} \)
67 \( 1 - 6.48T + 67T^{2} \)
71 \( 1 + 1.07T + 71T^{2} \)
73 \( 1 - 12.2T + 73T^{2} \)
79 \( 1 + 16.6T + 79T^{2} \)
83 \( 1 + 10.2T + 83T^{2} \)
89 \( 1 + 3.55T + 89T^{2} \)
97 \( 1 - 2.15T + 97T^{2} \)
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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

−8.040253592504339419675766670914, −7.19007474145318405383048100592, −6.57525726675808444758314593958, −5.79979720847023180287306262463, −5.16395629554868835693673887212, −4.39578526635401880891184353022, −3.64963928932096877481729866531, −2.39479008083085728310765330939, −1.08215593477978839633564662353, 0, 1.08215593477978839633564662353, 2.39479008083085728310765330939, 3.64963928932096877481729866531, 4.39578526635401880891184353022, 5.16395629554868835693673887212, 5.79979720847023180287306262463, 6.57525726675808444758314593958, 7.19007474145318405383048100592, 8.040253592504339419675766670914

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