Properties

Label 2-1904-1.1-c1-0-8
Degree $2$
Conductor $1904$
Sign $1$
Analytic cond. $15.2035$
Root an. cond. $3.89916$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.618·3-s − 1.61·5-s + 7-s − 2.61·9-s + 5.23·11-s − 3.23·13-s + 1.00·15-s + 17-s − 0.472·19-s − 0.618·21-s − 5.70·23-s − 2.38·25-s + 3.47·27-s + 7.70·29-s + 9.32·31-s − 3.23·33-s − 1.61·35-s − 8.47·37-s + 2.00·39-s + 11.0·41-s + 0.909·43-s + 4.23·45-s − 0.472·47-s + 49-s − 0.618·51-s − 13.7·53-s − 8.47·55-s + ⋯
L(s)  = 1  − 0.356·3-s − 0.723·5-s + 0.377·7-s − 0.872·9-s + 1.57·11-s − 0.897·13-s + 0.258·15-s + 0.242·17-s − 0.108·19-s − 0.134·21-s − 1.19·23-s − 0.476·25-s + 0.668·27-s + 1.43·29-s + 1.67·31-s − 0.563·33-s − 0.273·35-s − 1.39·37-s + 0.320·39-s + 1.73·41-s + 0.138·43-s + 0.631·45-s − 0.0688·47-s + 0.142·49-s − 0.0865·51-s − 1.89·53-s − 1.14·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1904\)    =    \(2^{4} \cdot 7 \cdot 17\)
Sign: $1$
Analytic conductor: \(15.2035\)
Root analytic conductor: \(3.89916\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1904,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.227024681\)
\(L(\frac12)\) \(\approx\) \(1.227024681\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - T \)
17 \( 1 - T \)
good3 \( 1 + 0.618T + 3T^{2} \)
5 \( 1 + 1.61T + 5T^{2} \)
11 \( 1 - 5.23T + 11T^{2} \)
13 \( 1 + 3.23T + 13T^{2} \)
19 \( 1 + 0.472T + 19T^{2} \)
23 \( 1 + 5.70T + 23T^{2} \)
29 \( 1 - 7.70T + 29T^{2} \)
31 \( 1 - 9.32T + 31T^{2} \)
37 \( 1 + 8.47T + 37T^{2} \)
41 \( 1 - 11.0T + 41T^{2} \)
43 \( 1 - 0.909T + 43T^{2} \)
47 \( 1 + 0.472T + 47T^{2} \)
53 \( 1 + 13.7T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 8.32T + 61T^{2} \)
67 \( 1 - 1.90T + 67T^{2} \)
71 \( 1 - 6.94T + 71T^{2} \)
73 \( 1 - 13.8T + 73T^{2} \)
79 \( 1 - 14.9T + 79T^{2} \)
83 \( 1 - 11.4T + 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 - 3.90T + 97T^{2} \)
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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

−9.209935207095168281070748925085, −8.258022563354538671012713496604, −7.85145214379333741567846646103, −6.66941540525329587574385984930, −6.19537584911371863983466434923, −5.06491180027596975030854899788, −4.31149189186211904874032482224, −3.44193973581641863921648003515, −2.23667949207185852046902020726, −0.75770666601970373686798374340, 0.75770666601970373686798374340, 2.23667949207185852046902020726, 3.44193973581641863921648003515, 4.31149189186211904874032482224, 5.06491180027596975030854899788, 6.19537584911371863983466434923, 6.66941540525329587574385984930, 7.85145214379333741567846646103, 8.258022563354538671012713496604, 9.209935207095168281070748925085

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