Properties

Label 2-1334-1.1-c1-0-14
Degree $2$
Conductor $1334$
Sign $1$
Analytic cond. $10.6520$
Root an. cond. $3.26374$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 4·7-s − 8-s − 3·9-s + 2·11-s + 6·13-s − 4·14-s + 16-s + 2·17-s + 3·18-s − 2·19-s − 2·22-s + 23-s − 5·25-s − 6·26-s + 4·28-s − 29-s − 32-s − 2·34-s − 3·36-s + 4·37-s + 2·38-s − 2·41-s + 10·43-s + 2·44-s − 46-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 1.51·7-s − 0.353·8-s − 9-s + 0.603·11-s + 1.66·13-s − 1.06·14-s + 1/4·16-s + 0.485·17-s + 0.707·18-s − 0.458·19-s − 0.426·22-s + 0.208·23-s − 25-s − 1.17·26-s + 0.755·28-s − 0.185·29-s − 0.176·32-s − 0.342·34-s − 1/2·36-s + 0.657·37-s + 0.324·38-s − 0.312·41-s + 1.52·43-s + 0.301·44-s − 0.147·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1334\)    =    \(2 \cdot 23 \cdot 29\)
Sign: $1$
Analytic conductor: \(10.6520\)
Root analytic conductor: \(3.26374\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1334} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1334,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.482570899\)
\(L(\frac12)\) \(\approx\) \(1.482570899\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
23 \( 1 - T \)
29 \( 1 + T \)
good3 \( 1 + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−9.362840693737650936052402691335, −8.737259013171114215932090957122, −8.136313333585105525919773653064, −7.51266302567345812935898598967, −6.18882132913877394849460983797, −5.72842264193310018375453685876, −4.47449759907179690290855698049, −3.44148842054825095067948308700, −2.07455635641233909533092691275, −1.06802436445112848630233397781, 1.06802436445112848630233397781, 2.07455635641233909533092691275, 3.44148842054825095067948308700, 4.47449759907179690290855698049, 5.72842264193310018375453685876, 6.18882132913877394849460983797, 7.51266302567345812935898598967, 8.136313333585105525919773653064, 8.737259013171114215932090957122, 9.362840693737650936052402691335

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