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Properties

Label 2-340-17.4-c1-0-5
Degree $2$
Conductor $340$
Sign $-0.125 + 0.992i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.26 − 1.26i)3-s + (0.707 − 0.707i)5-s + (−3.70 − 3.70i)7-s − 0.222i·9-s + (−3.73 − 3.73i)11-s + 2.63·13-s − 1.79i·15-s + (4.12 + 0.0118i)17-s + 4.31i·19-s − 9.39·21-s + (2.94 + 2.94i)23-s − 1.00i·25-s + (3.52 + 3.52i)27-s + (1.80 − 1.80i)29-s + (2.12 − 2.12i)31-s + ⋯
L(s)  = 1  + (0.732 − 0.732i)3-s + (0.316 − 0.316i)5-s + (−1.39 − 1.39i)7-s − 0.0742i·9-s + (−1.12 − 1.12i)11-s + 0.731·13-s − 0.463i·15-s + (0.999 + 0.00286i)17-s + 0.989i·19-s − 2.04·21-s + (0.614 + 0.614i)23-s − 0.200i·25-s + (0.678 + 0.678i)27-s + (0.334 − 0.334i)29-s + (0.382 − 0.382i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.125 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.125 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(340\)    =    \(2^{2} \cdot 5 \cdot 17\)
Sign: $-0.125 + 0.992i$
Analytic conductor: \(2.71491\)
Root analytic conductor: \(1.64769\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{340} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 340,\ (\ :1/2),\ -0.125 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.965220 - 1.09448i\)
\(L(\frac12)\) \(\approx\) \(0.965220 - 1.09448i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 + 0.707i)T \)
17 \( 1 + (-4.12 - 0.0118i)T \)
good3 \( 1 + (-1.26 + 1.26i)T - 3iT^{2} \)
7 \( 1 + (3.70 + 3.70i)T + 7iT^{2} \)
11 \( 1 + (3.73 + 3.73i)T + 11iT^{2} \)
13 \( 1 - 2.63T + 13T^{2} \)
19 \( 1 - 4.31iT - 19T^{2} \)
23 \( 1 + (-2.94 - 2.94i)T + 23iT^{2} \)
29 \( 1 + (-1.80 + 1.80i)T - 29iT^{2} \)
31 \( 1 + (-2.12 + 2.12i)T - 31iT^{2} \)
37 \( 1 + (-6.69 + 6.69i)T - 37iT^{2} \)
41 \( 1 + (-0.956 - 0.956i)T + 41iT^{2} \)
43 \( 1 - 0.613iT - 43T^{2} \)
47 \( 1 + 5.18T + 47T^{2} \)
53 \( 1 + 1.77iT - 53T^{2} \)
59 \( 1 + 7.97iT - 59T^{2} \)
61 \( 1 + (2.57 + 2.57i)T + 61iT^{2} \)
67 \( 1 - 11.7T + 67T^{2} \)
71 \( 1 + (5.08 - 5.08i)T - 71iT^{2} \)
73 \( 1 + (4.58 - 4.58i)T - 73iT^{2} \)
79 \( 1 + (-1.26 - 1.26i)T + 79iT^{2} \)
83 \( 1 - 12.8iT - 83T^{2} \)
89 \( 1 + 3.34T + 89T^{2} \)
97 \( 1 + (3.27 - 3.27i)T - 97iT^{2} \)
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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

−11.11306882215968833899333758295, −10.27963710836153313436837613571, −9.488925611411150932050029443515, −8.181870173043525026302160494794, −7.68642205661409764378057890804, −6.53920897530899094799321802083, −5.56766032603540251130336067516, −3.75894987790921083025193244757, −2.89134516898832585890231052238, −0.990845285173534557099795291489, 2.65128401700187413707885631520, 3.15426306030750436996835701625, 4.77567665017906543833483770052, 5.94258997236102453405363642714, 6.89455251488601225530578501029, 8.333905989276796792013865351767, 9.181431502629413493038864348094, 9.811028623792014829995911656456, 10.47258106859596018039329251299, 11.92804956276012162111366347345

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