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Properties

Label 2-77-7.6-c4-0-4
Degree $2$
Conductor $77$
Sign $-0.949 + 0.313i$
Analytic cond. $7.95948$
Root an. cond. $2.82125$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.518·2-s + 16.0i·3-s − 15.7·4-s + 43.3i·5-s − 8.33i·6-s + (46.5 − 15.3i)7-s + 16.4·8-s − 177.·9-s − 22.4i·10-s + 36.4·11-s − 253. i·12-s − 15.1i·13-s + (−24.1 + 7.96i)14-s − 697.·15-s + 243.·16-s − 377. i·17-s + ⋯
L(s)  = 1  − 0.129·2-s + 1.78i·3-s − 0.983·4-s + 1.73i·5-s − 0.231i·6-s + (0.949 − 0.313i)7-s + 0.256·8-s − 2.19·9-s − 0.224i·10-s + 0.301·11-s − 1.75i·12-s − 0.0897i·13-s + (−0.122 + 0.0406i)14-s − 3.09·15-s + 0.949·16-s − 1.30i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 + 0.313i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.949 + 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $-0.949 + 0.313i$
Analytic conductor: \(7.95948\)
Root analytic conductor: \(2.82125\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 77,\ (\ :2),\ -0.949 + 0.313i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.172068 - 1.06947i\)
\(L(\frac12)\) \(\approx\) \(0.172068 - 1.06947i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-46.5 + 15.3i)T \)
11 \( 1 - 36.4T \)
good2 \( 1 + 0.518T + 16T^{2} \)
3 \( 1 - 16.0iT - 81T^{2} \)
5 \( 1 - 43.3iT - 625T^{2} \)
13 \( 1 + 15.1iT - 2.85e4T^{2} \)
17 \( 1 + 377. iT - 8.35e4T^{2} \)
19 \( 1 - 597. iT - 1.30e5T^{2} \)
23 \( 1 - 438.T + 2.79e5T^{2} \)
29 \( 1 + 558.T + 7.07e5T^{2} \)
31 \( 1 - 523. iT - 9.23e5T^{2} \)
37 \( 1 + 3.57T + 1.87e6T^{2} \)
41 \( 1 + 468. iT - 2.82e6T^{2} \)
43 \( 1 - 1.49e3T + 3.41e6T^{2} \)
47 \( 1 - 1.21e3iT - 4.87e6T^{2} \)
53 \( 1 - 221.T + 7.89e6T^{2} \)
59 \( 1 + 1.59e3iT - 1.21e7T^{2} \)
61 \( 1 - 4.56e3iT - 1.38e7T^{2} \)
67 \( 1 + 4.46e3T + 2.01e7T^{2} \)
71 \( 1 + 7.62e3T + 2.54e7T^{2} \)
73 \( 1 - 2.84e3iT - 2.83e7T^{2} \)
79 \( 1 - 605.T + 3.89e7T^{2} \)
83 \( 1 + 5.60e3iT - 4.74e7T^{2} \)
89 \( 1 - 4.45e3iT - 6.27e7T^{2} \)
97 \( 1 - 9.41e3iT - 8.85e7T^{2} \)
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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

−14.47332946957456757967127184238, −13.95072281309513842367655887049, −11.61752263171172313238502630519, −10.67707254394685323830877949697, −10.06904157838275564966599414949, −9.009158912492003027737305781226, −7.57868538432871096017156196651, −5.57569866274947728782374441083, −4.30434132818034579549253470235, −3.24710743321326458558270309051, 0.63827778572380274915807373284, 1.65788777281723883628219306507, 4.63509225052282319467717820825, 5.76074023558775213890417820788, 7.59362046177258323014464754428, 8.570872555329241433088142366253, 9.021247992070111088322117713529, 11.37867197342525309212357754420, 12.46343672150995646106962915166, 13.03785083112087976351293561494

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