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Properties

Label 2-680-17.8-c1-0-16
Degree $2$
Conductor $680$
Sign $-0.617 + 0.786i$
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  + (1.01 − 2.43i)3-s + (−0.923 − 0.382i)5-s + (1.54 − 0.638i)7-s + (−2.80 − 2.80i)9-s + (0.182 + 0.439i)11-s − 3.35i·13-s + (−1.86 + 1.86i)15-s + (1.41 − 3.87i)17-s + (−2.12 + 2.12i)19-s − 4.40i·21-s + (−1.73 − 4.19i)23-s + (0.707 + 0.707i)25-s + (−2.35 + 0.976i)27-s + (−2.49 − 1.03i)29-s + (−2.29 + 5.55i)31-s + ⋯
L(s)  = 1  + (0.583 − 1.40i)3-s + (−0.413 − 0.171i)5-s + (0.582 − 0.241i)7-s + (−0.934 − 0.934i)9-s + (0.0548 + 0.132i)11-s − 0.930i·13-s + (−0.481 + 0.481i)15-s + (0.342 − 0.939i)17-s + (−0.486 + 0.486i)19-s − 0.960i·21-s + (−0.362 − 0.874i)23-s + (0.141 + 0.141i)25-s + (−0.453 + 0.187i)27-s + (−0.463 − 0.191i)29-s + (−0.412 + 0.996i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.617 + 0.786i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.617 + 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $-0.617 + 0.786i$
Analytic conductor: \(5.42982\)
Root analytic conductor: \(2.33019\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{680} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 680,\ (\ :1/2),\ -0.617 + 0.786i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.719603 - 1.48082i\)
\(L(\frac12)\) \(\approx\) \(0.719603 - 1.48082i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.41 + 3.87i)T \)
good3 \( 1 + (-1.01 + 2.43i)T + (-2.12 - 2.12i)T^{2} \)
7 \( 1 + (-1.54 + 0.638i)T + (4.94 - 4.94i)T^{2} \)
11 \( 1 + (-0.182 - 0.439i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + 3.35iT - 13T^{2} \)
19 \( 1 + (2.12 - 2.12i)T - 19iT^{2} \)
23 \( 1 + (1.73 + 4.19i)T + (-16.2 + 16.2i)T^{2} \)
29 \( 1 + (2.49 + 1.03i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (2.29 - 5.55i)T + (-21.9 - 21.9i)T^{2} \)
37 \( 1 + (1.54 - 3.74i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-10.8 + 4.49i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (2.89 + 2.89i)T + 43iT^{2} \)
47 \( 1 - 3.77iT - 47T^{2} \)
53 \( 1 + (-1.24 + 1.24i)T - 53iT^{2} \)
59 \( 1 + (-2.74 - 2.74i)T + 59iT^{2} \)
61 \( 1 + (2.91 - 1.20i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + 2.24T + 67T^{2} \)
71 \( 1 + (1.34 - 3.25i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-12.5 - 5.21i)T + (51.6 + 51.6i)T^{2} \)
79 \( 1 + (-1.81 - 4.37i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (-11.1 + 11.1i)T - 83iT^{2} \)
89 \( 1 + 11.4iT - 89T^{2} \)
97 \( 1 + (-12.7 - 5.26i)T + (68.5 + 68.5i)T^{2} \)
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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

−10.24591824353003208801523326593, −8.995442519311333387044680154224, −8.189021486772909319839075855939, −7.64754635651639029097612944846, −6.93919670772009278584356861647, −5.83099437000233348246929992491, −4.63719042361382833739107718521, −3.27592521197660061455345551707, −2.12750998697726256110496203024, −0.842626037946823451646732656359, 2.10135357776220885226617708948, 3.52795170617737911572453537667, 4.16467202794906481623997243175, 5.07779144700559762780498917436, 6.21882808930689482603889587783, 7.57082300402867252597199262139, 8.376097331041949313308710736020, 9.187769259544244692284400445208, 9.778692046571237365789813410860, 10.88557337067668726246637426019

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