↑

Properties

Label 2-2312-17.16-c1-0-25
Degree $2$
Conductor $2312$
Sign $0.443 - 0.896i$
Analytic cond. $18.4614$
Root an. cond. $4.29667$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.61i·3-s − 2.93i·5-s − 1.08i·7-s − 3.82·9-s + 4.14i·11-s − 2.58·13-s + 7.65·15-s + 7.65·19-s + 2.82·21-s − 6.30i·23-s − 3.58·25-s − 2.16i·27-s + 5.54i·29-s + 1.08i·31-s − 10.8·33-s + ⋯
L(s)  = 1  + 1.50i·3-s − 1.31i·5-s − 0.409i·7-s − 1.27·9-s + 1.24i·11-s − 0.717·13-s + 1.97·15-s + 1.75·19-s + 0.617·21-s − 1.31i·23-s − 0.717·25-s − 0.416i·27-s + 1.02i·29-s + 0.194i·31-s − 1.88·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2312\)    =    \(2^{3} \cdot 17^{2}\)
Sign: $0.443 - 0.896i$
Analytic conductor: \(18.4614\)
Root analytic conductor: \(4.29667\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2312} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2312,\ (\ :1/2),\ 0.443 - 0.896i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.698347515\)
\(L(\frac12)\) \(\approx\) \(1.698347515\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 \)
good3 \( 1 - 2.61iT - 3T^{2} \)
5 \( 1 + 2.93iT - 5T^{2} \)
7 \( 1 + 1.08iT - 7T^{2} \)
11 \( 1 - 4.14iT - 11T^{2} \)
13 \( 1 + 2.58T + 13T^{2} \)
19 \( 1 - 7.65T + 19T^{2} \)
23 \( 1 + 6.30iT - 23T^{2} \)
29 \( 1 - 5.54iT - 29T^{2} \)
31 \( 1 - 1.08iT - 31T^{2} \)
37 \( 1 + 10.7iT - 37T^{2} \)
41 \( 1 - 9.23iT - 41T^{2} \)
43 \( 1 - 0.343T + 43T^{2} \)
47 \( 1 - 1.17T + 47T^{2} \)
53 \( 1 - 8.24T + 53T^{2} \)
59 \( 1 - 7.17T + 59T^{2} \)
61 \( 1 - 4.46iT - 61T^{2} \)
67 \( 1 - 2.82T + 67T^{2} \)
71 \( 1 - 9.10iT - 71T^{2} \)
73 \( 1 - 7.97iT - 73T^{2} \)
79 \( 1 - 4.14iT - 79T^{2} \)
83 \( 1 - 12.8T + 83T^{2} \)
89 \( 1 - 4.24T + 89T^{2} \)
97 \( 1 - 4.46iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.198784346508878367546087619728, −8.733938745904681086339278594814, −7.65647697894518784049329530727, −6.96049404849932768583427308959, −5.47018013440779138438522380363, −5.03435815871631325645008382067, −4.43209278248891699681410868786, −3.75589194908446113673397065886, −2.49447428765089044172460181544, −0.974082015918101898379781050011, 0.75038540243419911384943993266, 2.05195723440276526321020835989, 2.89054880117051963058193205539, 3.53153333789225879127156383909, 5.28845331506140111419854918797, 5.94778822699684458888009490042, 6.62813042899600055525929630209, 7.43414186231207597141249221750, 7.70140830960490239736351374599, 8.674589679458866305499548110760

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