Properties

Label 2-1840-5.4-c1-0-20
Degree $2$
Conductor $1840$
Sign $0.757 - 0.653i$
Analytic cond. $14.6924$
Root an. cond. $3.83307$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.78i·3-s + (1.46 + 1.69i)5-s + 1.75i·7-s − 0.192·9-s + 4.77·11-s + 1.72i·13-s + (3.02 − 2.61i)15-s + 7.81i·17-s − 2.43·19-s + 3.13·21-s + i·23-s + (−0.731 + 4.94i)25-s − 5.01i·27-s − 7.86·29-s − 6.14·31-s + ⋯
L(s)  = 1  − 1.03i·3-s + (0.653 + 0.757i)5-s + 0.663i·7-s − 0.0640·9-s + 1.43·11-s + 0.479i·13-s + (0.780 − 0.673i)15-s + 1.89i·17-s − 0.559·19-s + 0.684·21-s + 0.208i·23-s + (−0.146 + 0.989i)25-s − 0.965i·27-s − 1.46·29-s − 1.10·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 - 0.653i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.757 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.757 - 0.653i$
Analytic conductor: \(14.6924\)
Root analytic conductor: \(3.83307\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1840,\ (\ :1/2),\ 0.757 - 0.653i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.998754204\)
\(L(\frac12)\) \(\approx\) \(1.998754204\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-1.46 - 1.69i)T \)
23 \( 1 - iT \)
good3 \( 1 + 1.78iT - 3T^{2} \)
7 \( 1 - 1.75iT - 7T^{2} \)
11 \( 1 - 4.77T + 11T^{2} \)
13 \( 1 - 1.72iT - 13T^{2} \)
17 \( 1 - 7.81iT - 17T^{2} \)
19 \( 1 + 2.43T + 19T^{2} \)
29 \( 1 + 7.86T + 29T^{2} \)
31 \( 1 + 6.14T + 31T^{2} \)
37 \( 1 - 6.83iT - 37T^{2} \)
41 \( 1 + 2.50T + 41T^{2} \)
43 \( 1 - 3.26iT - 43T^{2} \)
47 \( 1 - 8.46iT - 47T^{2} \)
53 \( 1 + 2.76iT - 53T^{2} \)
59 \( 1 - 1.91T + 59T^{2} \)
61 \( 1 + 3.50T + 61T^{2} \)
67 \( 1 + 12.7iT - 67T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 - 0.0111iT - 73T^{2} \)
79 \( 1 + 16.6T + 79T^{2} \)
83 \( 1 - 2.64iT - 83T^{2} \)
89 \( 1 - 13.1T + 89T^{2} \)
97 \( 1 + 15.8iT - 97T^{2} \)
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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

−9.277407761810918041413856622389, −8.570982315515976122746064463697, −7.62701069748278414787067573994, −6.79987430707157667755836450449, −6.26544024546337834734547117465, −5.76363004717439664253608378716, −4.25692192971509517664410903549, −3.35491292865003803807588634547, −1.87329077680714895846456021922, −1.68947733212085388026553443830, 0.74721543601711695574083415855, 2.05696802685984572610530490511, 3.58824144413364086282741380640, 4.16292126941723955662025525751, 5.03471109490484629857438838902, 5.68297631931882013516684817195, 6.86021051225909715615914815433, 7.47173208545879625589045292853, 8.901588565547597800453504743396, 9.126357771938035008120442348661

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