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Properties

Label 2-68-17.16-c1-0-1
Degree $2$
Conductor $68$
Sign $0.727 + 0.685i$
Analytic cond. $0.542982$
Root an. cond. $0.736873$
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  − 1.41i·3-s − 2.82i·5-s + 4.24i·7-s + 0.999·9-s + 1.41i·11-s − 4·13-s − 4.00·15-s + (3 + 2.82i)17-s − 4·19-s + 6·21-s + 1.41i·23-s − 3.00·25-s − 5.65i·27-s − 2.82i·29-s − 4.24i·31-s + ⋯
L(s)  = 1  − 0.816i·3-s − 1.26i·5-s + 1.60i·7-s + 0.333·9-s + 0.426i·11-s − 1.10·13-s − 1.03·15-s + (0.727 + 0.685i)17-s − 0.917·19-s + 1.30·21-s + 0.294i·23-s − 0.600·25-s − 1.08i·27-s − 0.525i·29-s − 0.762i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.685i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.727 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(68\)    =    \(2^{2} \cdot 17\)
Sign: $0.727 + 0.685i$
Analytic conductor: \(0.542982\)
Root analytic conductor: \(0.736873\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{68} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 68,\ (\ :1/2),\ 0.727 + 0.685i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.845995 - 0.335926i\)
\(L(\frac12)\) \(\approx\) \(0.845995 - 0.335926i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 + (-3 - 2.82i)T \)
good3 \( 1 + 1.41iT - 3T^{2} \)
5 \( 1 + 2.82iT - 5T^{2} \)
7 \( 1 - 4.24iT - 7T^{2} \)
11 \( 1 - 1.41iT - 11T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 - 1.41iT - 23T^{2} \)
29 \( 1 + 2.82iT - 29T^{2} \)
31 \( 1 + 4.24iT - 31T^{2} \)
37 \( 1 - 8.48iT - 37T^{2} \)
41 \( 1 + 11.3iT - 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + 12T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 8.48iT - 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 7.07iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 4.24iT - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 12T + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−14.78314990195680939291483258019, −13.12990486843229870070419757162, −12.42812820711201065307788952263, −11.96651015706018560929811900846, −9.859947069031234632947769625138, −8.775324792952608305234038274451, −7.73194344000881192909461523211, −6.06047072166510840925822598058, −4.78095038515256975486634747518, −2.02390448372635546016211511373, 3.31155621017059602303168202463, 4.64030908745122342322785253550, 6.73888792302150141204335430044, 7.60550254480635953750503585943, 9.668885494194043070932761404458, 10.45018406564663794713022036780, 11.09976168376602818722871050760, 12.83958162474705741517128327771, 14.29845803941890501871180999532, 14.61407234852838053605671649478

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