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Properties

Label 2-867-51.38-c0-0-0
Degree $2$
Conductor $867$
Sign $0.788 + 0.615i$
Analytic cond. $0.432689$
Root an. cond. $0.657791$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)3-s − 4-s + (−0.707 − 0.707i)7-s − 1.00i·9-s + (0.707 − 0.707i)12-s + 13-s + 16-s − i·19-s + 1.00·21-s − i·25-s + (0.707 + 0.707i)27-s + (0.707 + 0.707i)28-s + (0.707 − 0.707i)31-s + 1.00i·36-s + (0.707 − 0.707i)37-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)3-s − 4-s + (−0.707 − 0.707i)7-s − 1.00i·9-s + (0.707 − 0.707i)12-s + 13-s + 16-s − i·19-s + 1.00·21-s − i·25-s + (0.707 + 0.707i)27-s + (0.707 + 0.707i)28-s + (0.707 − 0.707i)31-s + 1.00i·36-s + (0.707 − 0.707i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $0.788 + 0.615i$
Analytic conductor: \(0.432689\)
Root analytic conductor: \(0.657791\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{867} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 867,\ (\ :0),\ 0.788 + 0.615i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5341931754\)
\(L(\frac12)\) \(\approx\) \(0.5341931754\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 \)
good2 \( 1 + T^{2} \)
5 \( 1 + iT^{2} \)
7 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
11 \( 1 - iT^{2} \)
13 \( 1 - T + T^{2} \)
19 \( 1 + iT - T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + iT^{2} \)
31 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
37 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 - iT - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
67 \( 1 + T + T^{2} \)
71 \( 1 + iT^{2} \)
73 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
79 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
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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

−10.24421910404071098771547226698, −9.504284741971350729448548427953, −8.881267554533344303647837741201, −7.82956995653123268778773059071, −6.55463420897253909556409351222, −5.92577237915978534975049110222, −4.74143429003153392351442594367, −4.12965944799308626802165964252, −3.20605174806432004589495505949, −0.69164488747293383763982657862, 1.36230105764151789874643651832, 3.06410198624547711167492618776, 4.24280479494134072127724901382, 5.46220063401616106482853365663, 5.93226144079959852517820556361, 6.90733675735922229648144695972, 8.058989724645424433171787735224, 8.677519439113841563643509808553, 9.616627686218561710206124139033, 10.40997805734827055461377401349

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