Properties

Label 2-31939-1.1-c1-0-1
Degree $2$
Conductor $31939$
Sign $-1$
Analytic cond. $255.034$
Root an. cond. $15.9697$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·4-s + 2·5-s − 2·7-s − 2·9-s + 2·11-s − 2·12-s − 5·13-s + 2·15-s + 4·16-s − 4·17-s − 19-s − 4·20-s − 2·21-s + 23-s − 25-s − 5·27-s + 4·28-s + 6·29-s + 8·31-s + 2·33-s − 4·35-s + 4·36-s + 8·37-s − 5·39-s − 43-s − 4·44-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s + 0.894·5-s − 0.755·7-s − 2/3·9-s + 0.603·11-s − 0.577·12-s − 1.38·13-s + 0.516·15-s + 16-s − 0.970·17-s − 0.229·19-s − 0.894·20-s − 0.436·21-s + 0.208·23-s − 1/5·25-s − 0.962·27-s + 0.755·28-s + 1.11·29-s + 1.43·31-s + 0.348·33-s − 0.676·35-s + 2/3·36-s + 1.31·37-s − 0.800·39-s − 0.152·43-s − 0.603·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 31939 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31939 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(31939\)    =    \(19 \cdot 41^{2}\)
Sign: $-1$
Analytic conductor: \(255.034\)
Root analytic conductor: \(15.9697\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{31939} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 31939,\ (\ :1/2),\ -1)\)

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)$
bad19 \( 1 + T \)
41 \( 1 \)
good2 \( 1 + p T^{2} \)
3 \( 1 - T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 - 13 T + p T^{2} \)
79 \( 1 + 7 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 3 T + p T^{2} \)
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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

−15.20823082237775, −14.57482332025906, −14.21917727094276, −13.79329276230633, −13.27395913140301, −12.94052493216806, −12.21766964767520, −11.77947365727413, −11.01594456300837, −10.11810046442617, −9.856164386009845, −9.456918241284461, −8.901669768974565, −8.458462529925675, −7.864981837662204, −7.061812229926881, −6.378168361802137, −5.979285311883991, −5.234764011635853, −4.569385135363611, −4.116521090652453, −3.155005880280376, −2.670811808908047, −2.064027630627257, −0.9071135523850221, 0, 0.9071135523850221, 2.064027630627257, 2.670811808908047, 3.155005880280376, 4.116521090652453, 4.569385135363611, 5.234764011635853, 5.979285311883991, 6.378168361802137, 7.061812229926881, 7.864981837662204, 8.458462529925675, 8.901669768974565, 9.456918241284461, 9.856164386009845, 10.11810046442617, 11.01594456300837, 11.77947365727413, 12.21766964767520, 12.94052493216806, 13.27395913140301, 13.79329276230633, 14.21917727094276, 14.57482332025906, 15.20823082237775

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