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Properties

Label 2-680-136.101-c1-0-67
Degree $2$
Conductor $680$
Sign $0.823 + 0.567i$
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  + (1.35 − 0.415i)2-s + 2.32·3-s + (1.65 − 1.12i)4-s + 5-s + (3.14 − 0.966i)6-s − 1.14i·7-s + (1.77 − 2.20i)8-s + 2.41·9-s + (1.35 − 0.415i)10-s − 5.72·11-s + (3.85 − 2.61i)12-s + 5.27i·13-s + (−0.474 − 1.54i)14-s + 2.32·15-s + (1.47 − 3.71i)16-s + (0.303 + 4.11i)17-s + ⋯
L(s)  = 1  + (0.955 − 0.293i)2-s + 1.34·3-s + (0.827 − 0.561i)4-s + 0.447·5-s + (1.28 − 0.394i)6-s − 0.431i·7-s + (0.626 − 0.779i)8-s + 0.805·9-s + (0.427 − 0.131i)10-s − 1.72·11-s + (1.11 − 0.754i)12-s + 1.46i·13-s + (−0.126 − 0.412i)14-s + 0.600·15-s + (0.369 − 0.929i)16-s + (0.0736 + 0.997i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 + 0.567i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.823 + 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $0.823 + 0.567i$
Analytic conductor: \(5.42982\)
Root analytic conductor: \(2.33019\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{680} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 680,\ (\ :1/2),\ 0.823 + 0.567i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.81806 - 1.18706i\)
\(L(\frac12)\) \(\approx\) \(3.81806 - 1.18706i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.35 + 0.415i)T \)
5 \( 1 - T \)
17 \( 1 + (-0.303 - 4.11i)T \)
good3 \( 1 - 2.32T + 3T^{2} \)
7 \( 1 + 1.14iT - 7T^{2} \)
11 \( 1 + 5.72T + 11T^{2} \)
13 \( 1 - 5.27iT - 13T^{2} \)
19 \( 1 + 7.03iT - 19T^{2} \)
23 \( 1 - 2.69iT - 23T^{2} \)
29 \( 1 + 6.20T + 29T^{2} \)
31 \( 1 - 5.46iT - 31T^{2} \)
37 \( 1 + 0.842T + 37T^{2} \)
41 \( 1 + 6.06iT - 41T^{2} \)
43 \( 1 - 2.67iT - 43T^{2} \)
47 \( 1 - 6.55T + 47T^{2} \)
53 \( 1 - 11.7iT - 53T^{2} \)
59 \( 1 - 0.687iT - 59T^{2} \)
61 \( 1 - 11.0T + 61T^{2} \)
67 \( 1 + 12.3iT - 67T^{2} \)
71 \( 1 + 6.57iT - 71T^{2} \)
73 \( 1 + 10.7iT - 73T^{2} \)
79 \( 1 + 0.186iT - 79T^{2} \)
83 \( 1 - 3.67iT - 83T^{2} \)
89 \( 1 + 6.77T + 89T^{2} \)
97 \( 1 - 4.14iT - 97T^{2} \)
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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

−10.55743105014506657649272695588, −9.527008691314608104091147782424, −8.803425184517420642297504793462, −7.60982303193735212136648334324, −6.99667942614763520703194530135, −5.72780464433789631879914264594, −4.70127424338633521981840464565, −3.69863788430279743646882006689, −2.66200964129014404541510600246, −1.88869498117889042397226990411, 2.30498333815325176899921159090, 2.80894666803030914375511514642, 3.83236411356177553481972310983, 5.37149587033279109782379225635, 5.67504774304618960983261218054, 7.26888795580401810078346802437, 7.987400057929130610311812096905, 8.437686243719419848237901077848, 9.798936614799755928027843235233, 10.42463664739029231478859940230

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