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Properties

Label 2-680-85.64-c1-0-1
Degree $2$
Conductor $680$
Sign $-0.497 - 0.867i$
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  + (0.513 − 0.513i)3-s + (−2.10 − 0.759i)5-s + (−0.542 − 0.542i)7-s + 2.47i·9-s + (−2.81 + 2.81i)11-s + 1.01i·13-s + (−1.46 + 0.689i)15-s + (−4.11 − 0.167i)17-s + 2.31i·19-s − 0.557·21-s + (−4.31 − 4.31i)23-s + (3.84 + 3.19i)25-s + (2.80 + 2.80i)27-s + (2.92 + 2.92i)29-s + (5.35 + 5.35i)31-s + ⋯
L(s)  = 1  + (0.296 − 0.296i)3-s + (−0.940 − 0.339i)5-s + (−0.205 − 0.205i)7-s + 0.824i·9-s + (−0.848 + 0.848i)11-s + 0.280i·13-s + (−0.379 + 0.178i)15-s + (−0.999 − 0.0405i)17-s + 0.530i·19-s − 0.121·21-s + (−0.898 − 0.898i)23-s + (0.769 + 0.639i)25-s + (0.540 + 0.540i)27-s + (0.543 + 0.543i)29-s + (0.960 + 0.960i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.497 - 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.497 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $-0.497 - 0.867i$
Analytic conductor: \(5.42982\)
Root analytic conductor: \(2.33019\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{680} (489, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 680,\ (\ :1/2),\ -0.497 - 0.867i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.277404 + 0.478882i\)
\(L(\frac12)\) \(\approx\) \(0.277404 + 0.478882i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.10 + 0.759i)T \)
17 \( 1 + (4.11 + 0.167i)T \)
good3 \( 1 + (-0.513 + 0.513i)T - 3iT^{2} \)
7 \( 1 + (0.542 + 0.542i)T + 7iT^{2} \)
11 \( 1 + (2.81 - 2.81i)T - 11iT^{2} \)
13 \( 1 - 1.01iT - 13T^{2} \)
19 \( 1 - 2.31iT - 19T^{2} \)
23 \( 1 + (4.31 + 4.31i)T + 23iT^{2} \)
29 \( 1 + (-2.92 - 2.92i)T + 29iT^{2} \)
31 \( 1 + (-5.35 - 5.35i)T + 31iT^{2} \)
37 \( 1 + (7.83 - 7.83i)T - 37iT^{2} \)
41 \( 1 + (6.53 - 6.53i)T - 41iT^{2} \)
43 \( 1 + 2.05T + 43T^{2} \)
47 \( 1 + 12.2iT - 47T^{2} \)
53 \( 1 - 6.90T + 53T^{2} \)
59 \( 1 - 6.13iT - 59T^{2} \)
61 \( 1 + (10.2 - 10.2i)T - 61iT^{2} \)
67 \( 1 + 8.92iT - 67T^{2} \)
71 \( 1 + (8.76 + 8.76i)T + 71iT^{2} \)
73 \( 1 + (6.18 - 6.18i)T - 73iT^{2} \)
79 \( 1 + (-10.0 + 10.0i)T - 79iT^{2} \)
83 \( 1 - 3.11T + 83T^{2} \)
89 \( 1 + 1.08T + 89T^{2} \)
97 \( 1 + (3.34 - 3.34i)T - 97iT^{2} \)
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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

−10.49240318646305597877717765895, −10.24170416427941141717432284696, −8.644835797443619225134729370859, −8.305741725130314062571760457935, −7.31433635937621104336190435827, −6.64777113806561174368416888988, −5.01909326226558672848381132570, −4.47185605477836491034685442837, −3.11081339757837768806558182553, −1.84781722015798992620636874100, 0.26963946133542833077879086453, 2.63469627469195338471932847195, 3.52556262644322038028722505430, 4.43208918553824820616115442251, 5.76050732082299708507376138245, 6.66983363621776890389600836367, 7.71940533471537285895235505992, 8.467073398280805958972475167222, 9.247464755110688203724941448986, 10.26800924937641179128307849665

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