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Properties

Label 2-680-17.15-c1-0-2
Degree $2$
Conductor $680$
Sign $-0.232 - 0.972i$
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  + (−0.251 − 0.606i)3-s + (−0.923 + 0.382i)5-s + (−3.28 − 1.36i)7-s + (1.81 − 1.81i)9-s + (−2.33 + 5.63i)11-s − 2.16i·13-s + (0.464 + 0.464i)15-s + (−0.338 + 4.10i)17-s + (5.28 + 5.28i)19-s + 2.33i·21-s + (−1.97 + 4.75i)23-s + (0.707 − 0.707i)25-s + (−3.37 − 1.39i)27-s + (−6.11 + 2.53i)29-s + (1.36 + 3.30i)31-s + ⋯
L(s)  = 1  + (−0.145 − 0.350i)3-s + (−0.413 + 0.171i)5-s + (−1.24 − 0.514i)7-s + (0.605 − 0.605i)9-s + (−0.704 + 1.70i)11-s − 0.600i·13-s + (0.119 + 0.119i)15-s + (−0.0820 + 0.996i)17-s + (1.21 + 1.21i)19-s + 0.509i·21-s + (−0.410 + 0.991i)23-s + (0.141 − 0.141i)25-s + (−0.650 − 0.269i)27-s + (−1.13 + 0.470i)29-s + (0.245 + 0.592i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.232 - 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.232 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $-0.232 - 0.972i$
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: \((2,\ 680,\ (\ :1/2),\ -0.232 - 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.377682 + 0.478566i\)
\(L(\frac12)\) \(\approx\) \(0.377682 + 0.478566i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.338 - 4.10i)T \)
good3 \( 1 + (0.251 + 0.606i)T + (-2.12 + 2.12i)T^{2} \)
7 \( 1 + (3.28 + 1.36i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (2.33 - 5.63i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + 2.16iT - 13T^{2} \)
19 \( 1 + (-5.28 - 5.28i)T + 19iT^{2} \)
23 \( 1 + (1.97 - 4.75i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (6.11 - 2.53i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (-1.36 - 3.30i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (1.31 + 3.17i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (1.93 + 0.802i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (4.83 - 4.83i)T - 43iT^{2} \)
47 \( 1 + 5.61iT - 47T^{2} \)
53 \( 1 + (-5.78 - 5.78i)T + 53iT^{2} \)
59 \( 1 + (8.43 - 8.43i)T - 59iT^{2} \)
61 \( 1 + (-2.78 - 1.15i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + 0.961T + 67T^{2} \)
71 \( 1 + (4.01 + 9.69i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (-0.933 + 0.386i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (-0.562 + 1.35i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (1.78 + 1.78i)T + 83iT^{2} \)
89 \( 1 - 10.8iT - 89T^{2} \)
97 \( 1 + (5.82 - 2.41i)T + (68.5 - 68.5i)T^{2} \)
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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

−10.30225589404563155418654315767, −10.12929711085757458099666385858, −9.238011137725989011112767079864, −7.66853357511683099131887000165, −7.39431341919497251238307919548, −6.44527570569237363380318181230, −5.42230937317841385391322729461, −4.01633353534308501751052242152, −3.31345315226495001212140482336, −1.60013272822527458258820054147, 0.32720504344023779500326678756, 2.62460328169028711537503689656, 3.51966798163784572855011778698, 4.79421082569333731397175542306, 5.64888613633628749738685858584, 6.67138759970949619748743574652, 7.61165688775759582856474035729, 8.648240740387700255933269567737, 9.421547906314233549381725054344, 10.14472473848940198605628757016

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