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Properties

Label 2-680-17.15-c1-0-9
Degree $2$
Conductor $680$
Sign $0.948 + 0.316i$
Analytic cond. $5.42982$
Root an. cond. $2.33019$
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  + (0.221 + 0.534i)3-s + (−0.923 + 0.382i)5-s + (0.202 + 0.0839i)7-s + (1.88 − 1.88i)9-s + (1.75 − 4.24i)11-s − 0.531i·13-s + (−0.409 − 0.409i)15-s + (−3.31 − 2.45i)17-s + (4.37 + 4.37i)19-s + 0.126i·21-s + (0.815 − 1.96i)23-s + (0.707 − 0.707i)25-s + (3.02 + 1.25i)27-s + (5.90 − 2.44i)29-s + (2.53 + 6.13i)31-s + ⋯
L(s)  = 1  + (0.127 + 0.308i)3-s + (−0.413 + 0.171i)5-s + (0.0766 + 0.0317i)7-s + (0.628 − 0.628i)9-s + (0.529 − 1.27i)11-s − 0.147i·13-s + (−0.105 − 0.105i)15-s + (−0.802 − 0.596i)17-s + (1.00 + 1.00i)19-s + 0.0277i·21-s + (0.170 − 0.410i)23-s + (0.141 − 0.141i)25-s + (0.583 + 0.241i)27-s + (1.09 − 0.454i)29-s + (0.456 + 1.10i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $0.948 + 0.316i$
Analytic conductor: \(5.42982\)
Root analytic conductor: \(2.33019\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{680} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 680,\ (\ :1/2),\ 0.948 + 0.316i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55222 - 0.252123i\)
\(L(\frac12)\) \(\approx\) \(1.55222 - 0.252123i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.923 - 0.382i)T \)
17 \( 1 + (3.31 + 2.45i)T \)
good3 \( 1 + (-0.221 - 0.534i)T + (-2.12 + 2.12i)T^{2} \)
7 \( 1 + (-0.202 - 0.0839i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (-1.75 + 4.24i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + 0.531iT - 13T^{2} \)
19 \( 1 + (-4.37 - 4.37i)T + 19iT^{2} \)
23 \( 1 + (-0.815 + 1.96i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-5.90 + 2.44i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (-2.53 - 6.13i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (-0.386 - 0.933i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-6.09 - 2.52i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (-7.94 + 7.94i)T - 43iT^{2} \)
47 \( 1 - 6.22iT - 47T^{2} \)
53 \( 1 + (4.60 + 4.60i)T + 53iT^{2} \)
59 \( 1 + (-5.22 + 5.22i)T - 59iT^{2} \)
61 \( 1 + (13.5 + 5.62i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + 1.62T + 67T^{2} \)
71 \( 1 + (-0.558 - 1.34i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (8.81 - 3.64i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (1.94 - 4.69i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (-7.08 - 7.08i)T + 83iT^{2} \)
89 \( 1 - 0.293iT - 89T^{2} \)
97 \( 1 + (17.5 - 7.24i)T + (68.5 - 68.5i)T^{2} \)
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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

−10.45216119074216295878930541039, −9.544405282379841283807900675771, −8.768160440266551164794243538513, −7.928177675918494033091861155819, −6.84659399200659212252892719167, −6.08690442964910165271285745680, −4.81307654200485647545727405838, −3.80451462989414024958147860227, −2.95093439112909534889607571783, −0.995231330189092818868148497899, 1.39349127238250999281830668968, 2.65648831677993430602448197498, 4.28811367339868666832787308309, 4.73774864616918600200533698502, 6.22682234334877247439225115596, 7.26459405839176027286521083661, 7.65623686649590807064902153582, 8.872874507444245942177326202179, 9.589237088378184509588646498622, 10.55332847344124909811038289985

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