Properties

Label 2-1856-1.1-c3-0-71
Degree $2$
Conductor $1856$
Sign $1$
Analytic cond. $109.507$
Root an. cond. $10.4645$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7·3-s − 5·5-s + 2·7-s + 22·9-s + 37·11-s − 27·13-s − 35·15-s + 24·17-s − 88·19-s + 14·21-s + 28·23-s − 100·25-s − 35·27-s + 29·29-s + 143·31-s + 259·33-s − 10·35-s + 360·37-s − 189·39-s + 386·41-s + 381·43-s − 110·45-s + 103·47-s − 339·49-s + 168·51-s + 431·53-s − 185·55-s + ⋯
L(s)  = 1  + 1.34·3-s − 0.447·5-s + 0.107·7-s + 0.814·9-s + 1.01·11-s − 0.576·13-s − 0.602·15-s + 0.342·17-s − 1.06·19-s + 0.145·21-s + 0.253·23-s − 4/5·25-s − 0.249·27-s + 0.185·29-s + 0.828·31-s + 1.36·33-s − 0.0482·35-s + 1.59·37-s − 0.776·39-s + 1.47·41-s + 1.35·43-s − 0.364·45-s + 0.319·47-s − 0.988·49-s + 0.461·51-s + 1.11·53-s − 0.453·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1856\)    =    \(2^{6} \cdot 29\)
Sign: $1$
Analytic conductor: \(109.507\)
Root analytic conductor: \(10.4645\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1856} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1856,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.539402962\)
\(L(\frac12)\) \(\approx\) \(3.539402962\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 - p T \)
good3 \( 1 - 7 T + p^{3} T^{2} \)
5 \( 1 + p T + p^{3} T^{2} \)
7 \( 1 - 2 T + p^{3} T^{2} \)
11 \( 1 - 37 T + p^{3} T^{2} \)
13 \( 1 + 27 T + p^{3} T^{2} \)
17 \( 1 - 24 T + p^{3} T^{2} \)
19 \( 1 + 88 T + p^{3} T^{2} \)
23 \( 1 - 28 T + p^{3} T^{2} \)
31 \( 1 - 143 T + p^{3} T^{2} \)
37 \( 1 - 360 T + p^{3} T^{2} \)
41 \( 1 - 386 T + p^{3} T^{2} \)
43 \( 1 - 381 T + p^{3} T^{2} \)
47 \( 1 - 103 T + p^{3} T^{2} \)
53 \( 1 - 431 T + p^{3} T^{2} \)
59 \( 1 - 288 T + p^{3} T^{2} \)
61 \( 1 - 840 T + p^{3} T^{2} \)
67 \( 1 + 180 T + p^{3} T^{2} \)
71 \( 1 + 706 T + p^{3} T^{2} \)
73 \( 1 - 716 T + p^{3} T^{2} \)
79 \( 1 + 931 T + p^{3} T^{2} \)
83 \( 1 - 1188 T + p^{3} T^{2} \)
89 \( 1 + 642 T + p^{3} T^{2} \)
97 \( 1 - 486 T + p^{3} T^{2} \)
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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

−8.870707852987378592638247938105, −8.086638611644483166040358116262, −7.59330996550778218133344256801, −6.67475446269757354770951093105, −5.74154068336114673510948000112, −4.32353605657413057126080181358, −3.97503325832271973647303837011, −2.84254233945346869397993539508, −2.12972351725539975655583383911, −0.827894620932734192572087134526, 0.827894620932734192572087134526, 2.12972351725539975655583383911, 2.84254233945346869397993539508, 3.97503325832271973647303837011, 4.32353605657413057126080181358, 5.74154068336114673510948000112, 6.67475446269757354770951093105, 7.59330996550778218133344256801, 8.086638611644483166040358116262, 8.870707852987378592638247938105

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