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Properties

Label 2-640-128.93-c1-0-12
Degree $2$
Conductor $640$
Sign $-0.983 - 0.179i$
Analytic cond. $5.11042$
Root an. cond. $2.26062$
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  + (0.995 + 1.00i)2-s + (−1.24 − 0.122i)3-s + (−0.0187 + 1.99i)4-s + (0.471 + 0.881i)5-s + (−1.11 − 1.36i)6-s + (0.0123 + 0.0622i)7-s + (−2.02 + 1.97i)8-s + (−1.41 − 0.281i)9-s + (−0.416 + 1.35i)10-s + (2.38 + 1.95i)11-s + (0.267 − 2.47i)12-s + (−1.45 − 0.776i)13-s + (−0.0502 + 0.0744i)14-s + (−0.477 − 1.15i)15-s + (−3.99 − 0.0751i)16-s + (−2.95 + 7.13i)17-s + ⋯
L(s)  = 1  + (0.703 + 0.710i)2-s + (−0.716 − 0.0705i)3-s + (−0.00939 + 0.999i)4-s + (0.210 + 0.394i)5-s + (−0.454 − 0.558i)6-s + (0.00468 + 0.0235i)7-s + (−0.716 + 0.697i)8-s + (−0.472 − 0.0939i)9-s + (−0.131 + 0.427i)10-s + (0.718 + 0.589i)11-s + (0.0772 − 0.715i)12-s + (−0.402 − 0.215i)13-s + (−0.0134 + 0.0198i)14-s + (−0.123 − 0.297i)15-s + (−0.999 − 0.0187i)16-s + (−0.717 + 1.73i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.179i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 - 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $-0.983 - 0.179i$
Analytic conductor: \(5.11042\)
Root analytic conductor: \(2.26062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 640,\ (\ :1/2),\ -0.983 - 0.179i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.105869 + 1.17223i\)
\(L(\frac12)\) \(\approx\) \(0.105869 + 1.17223i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.995 - 1.00i)T \)
5 \( 1 + (-0.471 - 0.881i)T \)
good3 \( 1 + (1.24 + 0.122i)T + (2.94 + 0.585i)T^{2} \)
7 \( 1 + (-0.0123 - 0.0622i)T + (-6.46 + 2.67i)T^{2} \)
11 \( 1 + (-2.38 - 1.95i)T + (2.14 + 10.7i)T^{2} \)
13 \( 1 + (1.45 + 0.776i)T + (7.22 + 10.8i)T^{2} \)
17 \( 1 + (2.95 - 7.13i)T + (-12.0 - 12.0i)T^{2} \)
19 \( 1 + (4.24 - 1.28i)T + (15.7 - 10.5i)T^{2} \)
23 \( 1 + (-4.63 + 3.09i)T + (8.80 - 21.2i)T^{2} \)
29 \( 1 + (4.51 + 5.49i)T + (-5.65 + 28.4i)T^{2} \)
31 \( 1 + (1.84 - 1.84i)T - 31iT^{2} \)
37 \( 1 + (2.83 - 9.33i)T + (-30.7 - 20.5i)T^{2} \)
41 \( 1 + (3.76 + 5.63i)T + (-15.6 + 37.8i)T^{2} \)
43 \( 1 + (1.56 - 0.154i)T + (42.1 - 8.38i)T^{2} \)
47 \( 1 + (-6.19 - 2.56i)T + (33.2 + 33.2i)T^{2} \)
53 \( 1 + (4.17 - 5.08i)T + (-10.3 - 51.9i)T^{2} \)
59 \( 1 + (-5.84 + 3.12i)T + (32.7 - 49.0i)T^{2} \)
61 \( 1 + (1.08 - 10.9i)T + (-59.8 - 11.9i)T^{2} \)
67 \( 1 + (-0.295 + 2.99i)T + (-65.7 - 13.0i)T^{2} \)
71 \( 1 + (-5.64 + 1.12i)T + (65.5 - 27.1i)T^{2} \)
73 \( 1 + (2.00 - 10.0i)T + (-67.4 - 27.9i)T^{2} \)
79 \( 1 + (-10.2 + 4.22i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (-4.01 - 13.2i)T + (-69.0 + 46.1i)T^{2} \)
89 \( 1 + (-3.48 - 2.33i)T + (34.0 + 82.2i)T^{2} \)
97 \( 1 + (-9.48 + 9.48i)T - 97iT^{2} \)
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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

−11.07612878902663457374390382298, −10.35864196354234352884297510191, −8.985921782079103178995484461340, −8.295791117471105555897397551846, −7.03816921903708229285219185136, −6.41791549944216904340466467353, −5.74350246579803664845612752576, −4.65369131430032017503711270836, −3.69131415230171854018252718690, −2.24140233132437251379198618928, 0.52908697089568074408424320089, 2.19959531778266378932409938796, 3.47795726891487586902119632231, 4.76747999398903149198553358225, 5.31512029913859905640226563312, 6.30241576850434343300335541958, 7.16007489878327500288869420946, 8.996837814435268665307782259517, 9.194834195403718695090038401249, 10.53033022144175345296488504768

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