↑

Properties

Label 2-340-85.57-c1-0-3
Degree $2$
Conductor $340$
Sign $0.517 - 0.855i$
Analytic cond. $2.71491$
Root an. cond. $1.64769$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.841 − 1.25i)3-s + (−0.704 + 2.12i)5-s + (−0.993 + 4.99i)7-s + (0.270 + 0.654i)9-s + (−0.0182 + 0.0919i)11-s + 1.69i·13-s + (2.07 + 2.67i)15-s + (−4.04 − 0.808i)17-s + (0.555 − 1.34i)19-s + (5.45 + 5.45i)21-s + (7.06 − 4.72i)23-s + (−4.00 − 2.99i)25-s + (5.50 + 1.09i)27-s + (1.26 − 1.89i)29-s + (−1.33 − 6.73i)31-s + ⋯
L(s)  = 1  + (0.485 − 0.726i)3-s + (−0.315 + 0.948i)5-s + (−0.375 + 1.88i)7-s + (0.0903 + 0.218i)9-s + (−0.00551 + 0.0277i)11-s + 0.470i·13-s + (0.536 + 0.689i)15-s + (−0.980 − 0.196i)17-s + (0.127 − 0.307i)19-s + (1.18 + 1.18i)21-s + (1.47 − 0.984i)23-s + (−0.801 − 0.598i)25-s + (1.05 + 0.210i)27-s + (0.234 − 0.351i)29-s + (−0.240 − 1.20i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.517 - 0.855i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.517 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(340\)    =    \(2^{2} \cdot 5 \cdot 17\)
Sign: $0.517 - 0.855i$
Analytic conductor: \(2.71491\)
Root analytic conductor: \(1.64769\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{340} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 340,\ (\ :1/2),\ 0.517 - 0.855i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17435 + 0.662577i\)
\(L(\frac12)\) \(\approx\) \(1.17435 + 0.662577i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.704 - 2.12i)T \)
17 \( 1 + (4.04 + 0.808i)T \)
good3 \( 1 + (-0.841 + 1.25i)T + (-1.14 - 2.77i)T^{2} \)
7 \( 1 + (0.993 - 4.99i)T + (-6.46 - 2.67i)T^{2} \)
11 \( 1 + (0.0182 - 0.0919i)T + (-10.1 - 4.20i)T^{2} \)
13 \( 1 - 1.69iT - 13T^{2} \)
19 \( 1 + (-0.555 + 1.34i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-7.06 + 4.72i)T + (8.80 - 21.2i)T^{2} \)
29 \( 1 + (-1.26 + 1.89i)T + (-11.0 - 26.7i)T^{2} \)
31 \( 1 + (1.33 + 6.73i)T + (-28.6 + 11.8i)T^{2} \)
37 \( 1 + (-7.49 - 5.00i)T + (14.1 + 34.1i)T^{2} \)
41 \( 1 + (-0.409 - 0.612i)T + (-15.6 + 37.8i)T^{2} \)
43 \( 1 + (3.51 - 8.47i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 - 9.58T + 47T^{2} \)
53 \( 1 + (8.55 - 3.54i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-8.64 + 3.58i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (1.11 - 0.746i)T + (23.3 - 56.3i)T^{2} \)
67 \( 1 + (1.45 - 1.45i)T - 67iT^{2} \)
71 \( 1 + (-7.11 + 1.41i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (0.164 + 0.826i)T + (-67.4 + 27.9i)T^{2} \)
79 \( 1 + (-5.67 - 1.12i)T + (72.9 + 30.2i)T^{2} \)
83 \( 1 + (-0.540 - 1.30i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-2.30 + 2.30i)T - 89iT^{2} \)
97 \( 1 + (2.23 + 11.2i)T + (-89.6 + 37.1i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.64747819930784664352557330225, −11.03427573990574202324148989048, −9.629978570144661259447768275461, −8.807807889248933910584385549732, −7.941078196473962414685865302362, −6.82560041116004255339835773935, −6.16557788842595488857849211827, −4.70028425958759565533688851371, −2.83378475552585769841044677178, −2.32490412148915080301098306435, 0.948133961060832701609519666398, 3.44365086587845046036708114736, 4.10939720738561566734496692979, 5.08910454371218651209569511465, 6.77104350901538237613015612017, 7.61727613704846313676622524396, 8.767495194372334455451290013012, 9.488932994665563635421584298000, 10.41355824218697082299187167635, 11.10995808954786757522671780109

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