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Properties

Label 2-78-13.8-c2-0-1
Degree $2$
Conductor $78$
Sign $0.208 - 0.977i$
Analytic cond. $2.12534$
Root an. cond. $1.45785$
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 + i)2-s − 1.73·3-s + 2i·4-s + (4.73 + 4.73i)5-s + (−1.73 − 1.73i)6-s + (−2.73 + 2.73i)7-s + (−2 + 2i)8-s + 2.99·9-s + 9.46i·10-s + (1.73 − 1.73i)11-s − 3.46i·12-s + (9.92 + 8.39i)13-s − 5.46·14-s + (−8.19 − 8.19i)15-s − 4·16-s − 29.3i·17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s − 0.577·3-s + 0.5i·4-s + (0.946 + 0.946i)5-s + (−0.288 − 0.288i)6-s + (−0.390 + 0.390i)7-s + (−0.250 + 0.250i)8-s + 0.333·9-s + 0.946i·10-s + (0.157 − 0.157i)11-s − 0.288i·12-s + (0.763 + 0.645i)13-s − 0.390·14-s + (−0.546 − 0.546i)15-s − 0.250·16-s − 1.72i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.208 - 0.977i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.208 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(78\)    =    \(2 \cdot 3 \cdot 13\)
Sign: $0.208 - 0.977i$
Analytic conductor: \(2.12534\)
Root analytic conductor: \(1.45785\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{78} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 78,\ (\ :1),\ 0.208 - 0.977i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.15959 + 0.938136i\)
\(L(\frac12)\) \(\approx\) \(1.15959 + 0.938136i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
3 \( 1 + 1.73T \)
13 \( 1 + (-9.92 - 8.39i)T \)
good5 \( 1 + (-4.73 - 4.73i)T + 25iT^{2} \)
7 \( 1 + (2.73 - 2.73i)T - 49iT^{2} \)
11 \( 1 + (-1.73 + 1.73i)T - 121iT^{2} \)
17 \( 1 + 29.3iT - 289T^{2} \)
19 \( 1 + (11.2 + 11.2i)T + 361iT^{2} \)
23 \( 1 + 29.3iT - 529T^{2} \)
29 \( 1 - 31.8T + 841T^{2} \)
31 \( 1 + (-26.9 - 26.9i)T + 961iT^{2} \)
37 \( 1 + (30.8 - 30.8i)T - 1.36e3iT^{2} \)
41 \( 1 + (14.4 + 14.4i)T + 1.68e3iT^{2} \)
43 \( 1 + 25.1iT - 1.84e3T^{2} \)
47 \( 1 + (41.1 - 41.1i)T - 2.20e3iT^{2} \)
53 \( 1 - 2.28T + 2.80e3T^{2} \)
59 \( 1 + (-54.6 + 54.6i)T - 3.48e3iT^{2} \)
61 \( 1 + 7.42T + 3.72e3T^{2} \)
67 \( 1 + (60.6 + 60.6i)T + 4.48e3iT^{2} \)
71 \( 1 + (-38.9 - 38.9i)T + 5.04e3iT^{2} \)
73 \( 1 + (-40.3 + 40.3i)T - 5.32e3iT^{2} \)
79 \( 1 + 148.T + 6.24e3T^{2} \)
83 \( 1 + (-73.7 - 73.7i)T + 6.88e3iT^{2} \)
89 \( 1 + (-25.5 + 25.5i)T - 7.92e3iT^{2} \)
97 \( 1 + (86.0 + 86.0i)T + 9.40e3iT^{2} \)
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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

−14.18680129188508364831788199579, −13.68399655594912083839522198106, −12.34142067248177176306182985121, −11.23685165397754361936617826364, −10.08645766468685540650190577675, −8.785679004485147129329902668562, −6.75734404286969207946185556152, −6.36675238337253509913894055915, −4.86881179575685435150385921268, −2.79912030090785898018306733881, 1.45783827172181421360418103857, 3.94445375271168561737640816706, 5.47242680179009150713002915964, 6.33178979939309660747212104869, 8.421586010150319416309272017544, 9.854517143384924155962377121949, 10.58269284567613699185558686928, 11.99786103849490567196600561916, 13.04393537031989837569149443006, 13.42914448748834306668721690157

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