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Properties

Label 2-1617-231.230-c0-0-1
Degree $2$
Conductor $1617$
Sign $0.0287 - 0.999i$
Analytic cond. $0.806988$
Root an. cond. $0.898325$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.923 + 0.382i)3-s − 4-s − 0.765·5-s + (0.707 − 0.707i)9-s − i·11-s + (0.923 − 0.382i)12-s + (0.707 − 0.292i)15-s + 16-s + 0.765·20-s + 1.41i·23-s − 0.414·25-s + (−0.382 + 0.923i)27-s + 1.84i·31-s + (0.382 + 0.923i)33-s + (−0.707 + 0.707i)36-s + ⋯
L(s)  = 1  + (−0.923 + 0.382i)3-s − 4-s − 0.765·5-s + (0.707 − 0.707i)9-s − i·11-s + (0.923 − 0.382i)12-s + (0.707 − 0.292i)15-s + 16-s + 0.765·20-s + 1.41i·23-s − 0.414·25-s + (−0.382 + 0.923i)27-s + 1.84i·31-s + (0.382 + 0.923i)33-s + (−0.707 + 0.707i)36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0287 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0287 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $0.0287 - 0.999i$
Analytic conductor: \(0.806988\)
Root analytic conductor: \(0.898325\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1617} (1616, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1617,\ (\ :0),\ 0.0287 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3887499209\)
\(L(\frac12)\) \(\approx\) \(0.3887499209\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.923 - 0.382i)T \)
7 \( 1 \)
11 \( 1 + iT \)
good2 \( 1 + T^{2} \)
5 \( 1 + 0.765T + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - 1.41iT - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - 1.84iT - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + 1.84T + T^{2} \)
53 \( 1 - 1.41iT - T^{2} \)
59 \( 1 - 1.84T + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - 1.41T + T^{2} \)
71 \( 1 - 1.41iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 1.84T + T^{2} \)
97 \( 1 + 0.765iT - T^{2} \)
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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

−9.820978031524265221756747760042, −9.019514769085977566553282114578, −8.291386369427570509423438943484, −7.45367839591812696349430426771, −6.42706309018405934714216679969, −5.48026967062427055818100624053, −4.94362110177131911016346573651, −3.86953552490479324165207520545, −3.39902315706664253320203289873, −1.10437227290806204187497446335, 0.43735557941759378130771139959, 2.08065525258289364462419761531, 3.76664764716997834574191464336, 4.49890174123687391122752666643, 5.11463027141038271491387021610, 6.15128146578794339590619443779, 6.99855490050043973904284278653, 7.87874932451449842789517734039, 8.367157412402567522953008060124, 9.630361610071184283620064416814

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