Properties

Label 2-76-76.11-c2-0-14
Degree $2$
Conductor $76$
Sign $0.548 + 0.836i$
Analytic cond. $2.07085$
Root an. cond. $1.43904$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.72 − 1.00i)2-s + (−0.443 − 0.255i)3-s + (1.97 − 3.47i)4-s + (1.99 − 3.44i)5-s + (−1.02 + 0.00336i)6-s + 9.66i·7-s + (−0.0788 − 7.99i)8-s + (−4.36 − 7.56i)9-s + (−0.0261 − 7.96i)10-s − 2.62i·11-s + (−1.76 + 1.03i)12-s + (11.8 + 20.4i)13-s + (9.72 + 16.7i)14-s + (−1.76 + 1.01i)15-s + (−8.18 − 13.7i)16-s + (−3.40 + 5.90i)17-s + ⋯
L(s)  = 1  + (0.864 − 0.502i)2-s + (−0.147 − 0.0852i)3-s + (0.494 − 0.869i)4-s + (0.398 − 0.689i)5-s + (−0.170 + 0.000560i)6-s + 1.38i·7-s + (−0.00985 − 0.999i)8-s + (−0.485 − 0.840i)9-s + (−0.00261 − 0.796i)10-s − 0.238i·11-s + (−0.147 + 0.0862i)12-s + (0.907 + 1.57i)13-s + (0.694 + 1.19i)14-s + (−0.117 + 0.0678i)15-s + (−0.511 − 0.859i)16-s + (−0.200 + 0.347i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.548 + 0.836i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.548 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.548 + 0.836i$
Analytic conductor: \(2.07085\)
Root analytic conductor: \(1.43904\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1),\ 0.548 + 0.836i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.68549 - 0.910182i\)
\(L(\frac12)\) \(\approx\) \(1.68549 - 0.910182i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.72 + 1.00i)T \)
19 \( 1 + (12.0 - 14.7i)T \)
good3 \( 1 + (0.443 + 0.255i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (-1.99 + 3.44i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 - 9.66iT - 49T^{2} \)
11 \( 1 + 2.62iT - 121T^{2} \)
13 \( 1 + (-11.8 - 20.4i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + (3.40 - 5.90i)T + (-144.5 - 250. i)T^{2} \)
23 \( 1 + (17.2 - 9.96i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-0.445 - 0.772i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + 59.6iT - 961T^{2} \)
37 \( 1 + 7.80T + 1.36e3T^{2} \)
41 \( 1 + (-15.9 + 27.6i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-6.09 - 3.51i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-47.7 + 27.5i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (33.4 + 57.9i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (51.3 + 29.6i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (1.23 + 2.14i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (36.4 - 21.0i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (-88.6 - 51.1i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (7.82 - 13.5i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (63.6 + 36.7i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 18.4iT - 6.88e3T^{2} \)
89 \( 1 + (35.4 + 61.4i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (39.9 - 69.2i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−13.98174413152698509696472959426, −12.86103245324301511305331153305, −11.98887902899606394269347379723, −11.27935961368321307418599040548, −9.507082249288150030909419075956, −8.739643125100842165132558304516, −6.29025849904789849689232252748, −5.66448399166021601307032316183, −3.94751074247075493983462202257, −1.95003953391095869711901079836, 2.96810552505165928324406690541, 4.59188139230340394784511146902, 6.03893720581943341251266376140, 7.19930386598173884214953197195, 8.297078946873835687190135154999, 10.61643943506580363225034878945, 10.81774366326151889911470387698, 12.60988352705261539348074054910, 13.69438684859501844690320048979, 14.11961079598560350437408174148

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