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Properties

Label 2-3468-17.4-c1-0-19
Degree $2$
Conductor $3468$
Sign $0.992 - 0.122i$
Analytic cond. $27.6921$
Root an. cond. $5.26233$
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  + (−0.707 + 0.707i)3-s + (0.707 − 0.707i)5-s + (−1.41 − 1.41i)7-s − 1.00i·9-s + (2.12 + 2.12i)11-s − 3·13-s + 1.00i·15-s − 3i·19-s + 2.00·21-s + (2.12 + 2.12i)23-s + 4i·25-s + (0.707 + 0.707i)27-s + (−4.24 + 4.24i)29-s + (1.41 − 1.41i)31-s − 3·33-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (0.316 − 0.316i)5-s + (−0.534 − 0.534i)7-s − 0.333i·9-s + (0.639 + 0.639i)11-s − 0.832·13-s + 0.258i·15-s − 0.688i·19-s + 0.436·21-s + (0.442 + 0.442i)23-s + 0.800i·25-s + (0.136 + 0.136i)27-s + (−0.787 + 0.787i)29-s + (0.254 − 0.254i)31-s − 0.522·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.122i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3468 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3468\)    =    \(2^{2} \cdot 3 \cdot 17^{2}\)
Sign: $0.992 - 0.122i$
Analytic conductor: \(27.6921\)
Root analytic conductor: \(5.26233\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3468} (3217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3468,\ (\ :1/2),\ 0.992 - 0.122i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.465309022\)
\(L(\frac12)\) \(\approx\) \(1.465309022\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 \)
good5 \( 1 + (-0.707 + 0.707i)T - 5iT^{2} \)
7 \( 1 + (1.41 + 1.41i)T + 7iT^{2} \)
11 \( 1 + (-2.12 - 2.12i)T + 11iT^{2} \)
13 \( 1 + 3T + 13T^{2} \)
19 \( 1 + 3iT - 19T^{2} \)
23 \( 1 + (-2.12 - 2.12i)T + 23iT^{2} \)
29 \( 1 + (4.24 - 4.24i)T - 29iT^{2} \)
31 \( 1 + (-1.41 + 1.41i)T - 31iT^{2} \)
37 \( 1 + (-7.07 + 7.07i)T - 37iT^{2} \)
41 \( 1 + (-2.12 - 2.12i)T + 41iT^{2} \)
43 \( 1 - iT - 43T^{2} \)
47 \( 1 + 2T + 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + (-7.07 - 7.07i)T + 61iT^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + (-5.65 + 5.65i)T - 71iT^{2} \)
73 \( 1 + (-9.89 + 9.89i)T - 73iT^{2} \)
79 \( 1 + (-8.48 - 8.48i)T + 79iT^{2} \)
83 \( 1 + 14iT - 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 - 97iT^{2} \)
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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

−8.932454903147553068414607416515, −7.61341642088629326465413565175, −7.14066853808666880713994106861, −6.37557002715116534270664248355, −5.49344165969749118426776746462, −4.79529276834302941196430684129, −4.05417054001499145291206532713, −3.16474297406584947870663799446, −1.97471704272185681123726638782, −0.72917896901376125359774202641, 0.72041923503923917518665356270, 2.08329767085935837631279292086, 2.84122831966060381953649192717, 3.86357150871005791703007016137, 4.89156850526186868702445343745, 5.76313548148989655724555745453, 6.35343186256213086871756208362, 6.84926639362209840333312340108, 7.87498243952020621677383656070, 8.462810344009668862357785201242

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