Properties

Label 2-7938-1.1-c1-0-152
Degree $2$
Conductor $7938$
Sign $-1$
Analytic cond. $63.3852$
Root an. cond. $7.96148$
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-s + 4-s + 3.44·5-s − 8-s − 3.44·10-s − 2·11-s + 4.89·13-s + 16-s + 2·17-s − 7.44·19-s + 3.44·20-s + 2·22-s + 23-s + 6.89·25-s − 4.89·26-s − 2.89·29-s − 6·31-s − 32-s − 2·34-s − 7.79·37-s + 7.44·38-s − 3.44·40-s − 9.79·41-s − 2.89·43-s − 2·44-s − 46-s − 9.79·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 1.54·5-s − 0.353·8-s − 1.09·10-s − 0.603·11-s + 1.35·13-s + 0.250·16-s + 0.485·17-s − 1.70·19-s + 0.771·20-s + 0.426·22-s + 0.208·23-s + 1.37·25-s − 0.960·26-s − 0.538·29-s − 1.07·31-s − 0.176·32-s − 0.342·34-s − 1.28·37-s + 1.20·38-s − 0.545·40-s − 1.53·41-s − 0.442·43-s − 0.301·44-s − 0.147·46-s − 1.42·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7938\)    =    \(2 \cdot 3^{4} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(63.3852\)
Root analytic conductor: \(7.96148\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7938,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 3.44T + 5T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 - 4.89T + 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 7.44T + 19T^{2} \)
23 \( 1 - T + 23T^{2} \)
29 \( 1 + 2.89T + 29T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 + 7.79T + 37T^{2} \)
41 \( 1 + 9.79T + 41T^{2} \)
43 \( 1 + 2.89T + 43T^{2} \)
47 \( 1 + 9.79T + 47T^{2} \)
53 \( 1 - 1.10T + 53T^{2} \)
59 \( 1 + 2T + 59T^{2} \)
61 \( 1 + 11.4T + 61T^{2} \)
67 \( 1 + 3.10T + 67T^{2} \)
71 \( 1 + 9.89T + 71T^{2} \)
73 \( 1 + 2.89T + 73T^{2} \)
79 \( 1 - 7.89T + 79T^{2} \)
83 \( 1 - 2T + 83T^{2} \)
89 \( 1 + 7.10T + 89T^{2} \)
97 \( 1 - 6.89T + 97T^{2} \)
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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

−7.54971107766653956070480491456, −6.65029855042273025578529124128, −6.23311800672535171381941753116, −5.58670672241762335966478457075, −4.92210242290622280731260502350, −3.70816674682945070201229758295, −2.92447412002799934950487610594, −1.82171791204364726840246623034, −1.59004598603295821190051766644, 0, 1.59004598603295821190051766644, 1.82171791204364726840246623034, 2.92447412002799934950487610594, 3.70816674682945070201229758295, 4.92210242290622280731260502350, 5.58670672241762335966478457075, 6.23311800672535171381941753116, 6.65029855042273025578529124128, 7.54971107766653956070480491456

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