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Properties

Label 2-3311-3311.1938-c0-0-0
Degree $2$
Conductor $3311$
Sign $0.0640 - 0.997i$
Analytic cond. $1.65240$
Root an. cond. $1.28545$
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  + (−0.267 + 0.279i)2-s + (0.0381 + 0.849i)4-s + (−0.104 + 0.994i)7-s + (−0.538 − 0.470i)8-s + (0.447 − 0.894i)9-s + (0.995 − 0.0896i)11-s + (−0.249 − 0.294i)14-s + (−0.571 + 0.0514i)16-s + (0.130 + 0.363i)18-s + (−0.240 + 0.302i)22-s + (0.438 + 1.11i)23-s + (0.999 − 0.0299i)25-s + (−0.849 − 0.0508i)28-s + (0.172 − 0.278i)29-s + (0.584 − 0.732i)32-s + ⋯
L(s)  = 1  + (−0.267 + 0.279i)2-s + (0.0381 + 0.849i)4-s + (−0.104 + 0.994i)7-s + (−0.538 − 0.470i)8-s + (0.447 − 0.894i)9-s + (0.995 − 0.0896i)11-s + (−0.249 − 0.294i)14-s + (−0.571 + 0.0514i)16-s + (0.130 + 0.363i)18-s + (−0.240 + 0.302i)22-s + (0.438 + 1.11i)23-s + (0.999 − 0.0299i)25-s + (−0.849 − 0.0508i)28-s + (0.172 − 0.278i)29-s + (0.584 − 0.732i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0640 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0640 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3311\)    =    \(7 \cdot 11 \cdot 43\)
Sign: $0.0640 - 0.997i$
Analytic conductor: \(1.65240\)
Root analytic conductor: \(1.28545\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3311} (1938, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3311,\ (\ :0),\ 0.0640 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.163346682\)
\(L(\frac12)\) \(\approx\) \(1.163346682\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.104 - 0.994i)T \)
11 \( 1 + (-0.995 + 0.0896i)T \)
43 \( 1 + (0.998 + 0.0598i)T \)
good2 \( 1 + (0.267 - 0.279i)T + (-0.0448 - 0.998i)T^{2} \)
3 \( 1 + (-0.447 + 0.894i)T^{2} \)
5 \( 1 + (-0.999 + 0.0299i)T^{2} \)
13 \( 1 + (-0.646 + 0.762i)T^{2} \)
17 \( 1 + (-0.337 + 0.941i)T^{2} \)
19 \( 1 + (-0.0149 + 0.999i)T^{2} \)
23 \( 1 + (-0.438 - 1.11i)T + (-0.733 + 0.680i)T^{2} \)
29 \( 1 + (-0.172 + 0.278i)T + (-0.447 - 0.894i)T^{2} \)
31 \( 1 + (0.842 - 0.538i)T^{2} \)
37 \( 1 + (-0.748 - 1.68i)T + (-0.669 + 0.743i)T^{2} \)
41 \( 1 + (0.936 - 0.351i)T^{2} \)
47 \( 1 + (-0.858 - 0.512i)T^{2} \)
53 \( 1 + (-1.73 + 0.970i)T + (0.525 - 0.850i)T^{2} \)
59 \( 1 + (-0.134 + 0.990i)T^{2} \)
61 \( 1 + (0.842 + 0.538i)T^{2} \)
67 \( 1 + (1.56 - 0.235i)T + (0.955 - 0.294i)T^{2} \)
71 \( 1 + (0.0888 - 1.48i)T + (-0.992 - 0.119i)T^{2} \)
73 \( 1 + (-0.971 + 0.237i)T^{2} \)
79 \( 1 + (0.0248 + 0.116i)T + (-0.913 + 0.406i)T^{2} \)
83 \( 1 + (0.887 - 0.460i)T^{2} \)
89 \( 1 + (-0.988 - 0.149i)T^{2} \)
97 \( 1 + (0.393 - 0.919i)T^{2} \)
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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

−8.859813899878277202938434797652, −8.456276385639504672122274378194, −7.46753350541845014253646813828, −6.69387894879919205879003367475, −6.31820743740654256026453243374, −5.25391335322839563041162557081, −4.21284206014437866516996378363, −3.43545160757465384933291437331, −2.72301904097828477783809125094, −1.31670373236893675873433089398, 0.889239620016045217468450834875, 1.79849190215475754223406823090, 2.86908742456459571533962099597, 4.14365183741559101594158752792, 4.67575703308752219030270213941, 5.58338479029877687162395607130, 6.57148792947111646026452280685, 7.01487674497241432886262953152, 7.88214461383770375680773340760, 8.878256857730604926198422871891

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