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Properties

Label 2-680-680.509-c1-0-15
Degree $2$
Conductor $680$
Sign $-0.470 + 0.882i$
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  + (0.870 + 1.11i)2-s + 3.10i·3-s + (−0.485 + 1.94i)4-s + (−0.636 − 2.14i)5-s + (−3.46 + 2.70i)6-s − 1.38·7-s + (−2.58 + 1.14i)8-s − 6.66·9-s + (1.83 − 2.57i)10-s − 3.17·11-s + (−6.03 − 1.50i)12-s + 4.69·13-s + (−1.20 − 1.54i)14-s + (6.66 − 1.97i)15-s + (−3.52 − 1.88i)16-s + (−1.50 − 3.83i)17-s + ⋯
L(s)  = 1  + (0.615 + 0.788i)2-s + 1.79i·3-s + (−0.242 + 0.970i)4-s + (−0.284 − 0.958i)5-s + (−1.41 + 1.10i)6-s − 0.522·7-s + (−0.913 + 0.405i)8-s − 2.22·9-s + (0.580 − 0.814i)10-s − 0.955·11-s + (−1.74 − 0.435i)12-s + 1.30·13-s + (−0.321 − 0.412i)14-s + (1.72 − 0.511i)15-s + (−0.882 − 0.470i)16-s + (−0.366 − 0.930i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.470 + 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.470 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $-0.470 + 0.882i$
Analytic conductor: \(5.42982\)
Root analytic conductor: \(2.33019\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{680} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 680,\ (\ :1/2),\ -0.470 + 0.882i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.523563 - 0.872056i\)
\(L(\frac12)\) \(\approx\) \(0.523563 - 0.872056i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.870 - 1.11i)T \)
5 \( 1 + (0.636 + 2.14i)T \)
17 \( 1 + (1.50 + 3.83i)T \)
good3 \( 1 - 3.10iT - 3T^{2} \)
7 \( 1 + 1.38T + 7T^{2} \)
11 \( 1 + 3.17T + 11T^{2} \)
13 \( 1 - 4.69T + 13T^{2} \)
19 \( 1 - 7.70iT - 19T^{2} \)
23 \( 1 + 4.95T + 23T^{2} \)
29 \( 1 - 1.71T + 29T^{2} \)
31 \( 1 - 6.13iT - 31T^{2} \)
37 \( 1 - 3.98iT - 37T^{2} \)
41 \( 1 - 4.75iT - 41T^{2} \)
43 \( 1 - 10.3T + 43T^{2} \)
47 \( 1 - 5.75iT - 47T^{2} \)
53 \( 1 - 0.856T + 53T^{2} \)
59 \( 1 - 3.40iT - 59T^{2} \)
61 \( 1 - 4.33T + 61T^{2} \)
67 \( 1 + 7.67T + 67T^{2} \)
71 \( 1 + 12.8iT - 71T^{2} \)
73 \( 1 - 8.29T + 73T^{2} \)
79 \( 1 - 9.37iT - 79T^{2} \)
83 \( 1 + 5.94T + 83T^{2} \)
89 \( 1 + 8.65T + 89T^{2} \)
97 \( 1 - 10.7T + 97T^{2} \)
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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

−10.99835970534925981091244082561, −10.04321615336208759193942789752, −9.269267829680266252891029308135, −8.467481185100334433993837077834, −7.87965951959822340646367716768, −6.16662090070433589882293254733, −5.51109233926848874066326438361, −4.64813757961639439136193067305, −3.91620415789288077966804534731, −3.09562978810251767010777608829, 0.41835407385782625999515594567, 2.08930302262105357272310702226, 2.79763933657946812805941347050, 3.93794114902513349755068534262, 5.71608781896319079462833677253, 6.30264542523258341397034320235, 7.03591543939621035461482437766, 8.028947850385965873460611352605, 8.957514628194883591448980446739, 10.32434947612065491368276176564

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