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Properties

Label 2-680-680.349-c1-0-17
Degree $2$
Conductor $680$
Sign $0.482 - 0.875i$
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  + (−0.648 + 1.25i)2-s + (−1.86 − 0.773i)3-s + (−1.15 − 1.63i)4-s + (−1.01 + 1.99i)5-s + (2.18 − 1.84i)6-s + (−0.739 + 0.306i)7-s + (2.80 − 0.397i)8-s + (0.764 + 0.764i)9-s + (−1.84 − 2.56i)10-s + (−1.31 − 3.17i)11-s + (0.900 + 3.93i)12-s − 2.51i·13-s + (0.0949 − 1.12i)14-s + (3.43 − 2.93i)15-s + (−1.31 + 3.77i)16-s + (−0.897 + 4.02i)17-s + ⋯
L(s)  = 1  + (−0.458 + 0.888i)2-s + (−1.07 − 0.446i)3-s + (−0.579 − 0.815i)4-s + (−0.454 + 0.890i)5-s + (0.890 − 0.752i)6-s + (−0.279 + 0.115i)7-s + (0.990 − 0.140i)8-s + (0.254 + 0.254i)9-s + (−0.582 − 0.812i)10-s + (−0.396 − 0.957i)11-s + (0.260 + 1.13i)12-s − 0.698i·13-s + (0.0253 − 0.301i)14-s + (0.887 − 0.756i)15-s + (−0.329 + 0.944i)16-s + (−0.217 + 0.976i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.482 - 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.482 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $0.482 - 0.875i$
Analytic conductor: \(5.42982\)
Root analytic conductor: \(2.33019\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{680} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 680,\ (\ :1/2),\ 0.482 - 0.875i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.446011 + 0.263354i\)
\(L(\frac12)\) \(\approx\) \(0.446011 + 0.263354i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.648 - 1.25i)T \)
5 \( 1 + (1.01 - 1.99i)T \)
17 \( 1 + (0.897 - 4.02i)T \)
good3 \( 1 + (1.86 + 0.773i)T + (2.12 + 2.12i)T^{2} \)
7 \( 1 + (0.739 - 0.306i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (1.31 + 3.17i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + 2.51iT - 13T^{2} \)
19 \( 1 + (0.658 + 0.658i)T + 19iT^{2} \)
23 \( 1 + (1.90 + 4.59i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (-9.10 - 3.77i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (-3.43 - 1.42i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (1.11 + 0.462i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-4.76 - 11.5i)T + (-28.9 + 28.9i)T^{2} \)
43 \( 1 + (-2.63 - 2.63i)T + 43iT^{2} \)
47 \( 1 - 10.2T + 47T^{2} \)
53 \( 1 + (3.65 - 3.65i)T - 53iT^{2} \)
59 \( 1 + (-5.24 + 5.24i)T - 59iT^{2} \)
61 \( 1 + (-8.77 + 3.63i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + 5.11T + 67T^{2} \)
71 \( 1 + (-7.58 - 3.14i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (-5.76 - 2.38i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (11.6 - 4.83i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (-4.85 + 4.85i)T - 83iT^{2} \)
89 \( 1 - 1.79iT - 89T^{2} \)
97 \( 1 + (5.86 + 2.43i)T + (68.5 + 68.5i)T^{2} \)
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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

−10.74961828650942553716023735025, −9.950139704473252454471169770634, −8.518216967286083693471955342651, −8.023915339222877704171439873551, −6.83700575302960681788474821043, −6.30692195898180166747574929515, −5.70444804754302817188095571518, −4.47035978271618873767352348818, −2.94970157836518231109070536521, −0.75425152056169607684489028044, 0.61122853516714278123419225668, 2.30667650946948260741688258332, 4.00345672588453283882858625051, 4.64631350862163368783761792007, 5.47235906824367493838509710301, 6.94781456934113206932770681321, 7.88392076182712323722573230225, 8.888159130829489868155983771625, 9.715392547642969997385895030233, 10.32136183531115253638961541487

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