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Properties

Label 2-39-13.5-c2-0-1
Degree $2$
Conductor $39$
Sign $0.749 - 0.662i$
Analytic cond. $1.06267$
Root an. cond. $1.03086$
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  + (−0.676 + 0.676i)2-s + 1.73·3-s + 3.08i·4-s + (1.58 − 1.58i)5-s + (−1.17 + 1.17i)6-s + (3.96 + 3.96i)7-s + (−4.79 − 4.79i)8-s + 2.99·9-s + 2.13i·10-s + (−10.0 − 10.0i)11-s + 5.34i·12-s + (−5.41 − 11.8i)13-s − 5.36·14-s + (2.73 − 2.73i)15-s − 5.85·16-s − 6.40i·17-s + ⋯
L(s)  = 1  + (−0.338 + 0.338i)2-s + 0.577·3-s + 0.771i·4-s + (0.316 − 0.316i)5-s + (−0.195 + 0.195i)6-s + (0.566 + 0.566i)7-s + (−0.599 − 0.599i)8-s + 0.333·9-s + 0.213i·10-s + (−0.912 − 0.912i)11-s + 0.445i·12-s + (−0.416 − 0.908i)13-s − 0.383·14-s + (0.182 − 0.182i)15-s − 0.366·16-s − 0.376i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.749 - 0.662i$
Analytic conductor: \(1.06267\)
Root analytic conductor: \(1.03086\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :1),\ 0.749 - 0.662i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.00817 + 0.381742i\)
\(L(\frac12)\) \(\approx\) \(1.00817 + 0.381742i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.73T \)
13 \( 1 + (5.41 + 11.8i)T \)
good2 \( 1 + (0.676 - 0.676i)T - 4iT^{2} \)
5 \( 1 + (-1.58 + 1.58i)T - 25iT^{2} \)
7 \( 1 + (-3.96 - 3.96i)T + 49iT^{2} \)
11 \( 1 + (10.0 + 10.0i)T + 121iT^{2} \)
17 \( 1 + 6.40iT - 289T^{2} \)
19 \( 1 + (-7.73 + 7.73i)T - 361iT^{2} \)
23 \( 1 - 34.8iT - 529T^{2} \)
29 \( 1 - 8.25T + 841T^{2} \)
31 \( 1 + (-13.8 + 13.8i)T - 961iT^{2} \)
37 \( 1 + (37.2 + 37.2i)T + 1.36e3iT^{2} \)
41 \( 1 + (28.7 - 28.7i)T - 1.68e3iT^{2} \)
43 \( 1 - 42.7iT - 1.84e3T^{2} \)
47 \( 1 + (-37.5 - 37.5i)T + 2.20e3iT^{2} \)
53 \( 1 + 96.7T + 2.80e3T^{2} \)
59 \( 1 + (-37.8 - 37.8i)T + 3.48e3iT^{2} \)
61 \( 1 - 91.2T + 3.72e3T^{2} \)
67 \( 1 + (42.4 - 42.4i)T - 4.48e3iT^{2} \)
71 \( 1 + (-29.8 + 29.8i)T - 5.04e3iT^{2} \)
73 \( 1 + (10.9 + 10.9i)T + 5.32e3iT^{2} \)
79 \( 1 - 102.T + 6.24e3T^{2} \)
83 \( 1 + (85.2 - 85.2i)T - 6.88e3iT^{2} \)
89 \( 1 + (63.8 + 63.8i)T + 7.92e3iT^{2} \)
97 \( 1 + (58.7 - 58.7i)T - 9.40e3iT^{2} \)
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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

−16.03794724308503595431442404428, −15.27630190960927127462360505635, −13.68664786890357135284021401603, −12.78917911429072617279854845818, −11.39327146635088726590594767136, −9.568617974312794498961734898440, −8.423608459973215656064075113426, −7.53784838689617265411190264529, −5.39103043234543743388837970450, −3.02295498421467578392680367546, 2.12801468983680167018523604111, 4.79585361892886897416092478165, 6.81870528862711777062223961694, 8.430421316134817823314269056778, 9.956163527017532331289856602510, 10.57281463283579905963948838184, 12.20105531349304757490028881345, 13.90733282433395227757753514667, 14.50853481715365895961816729312, 15.65029083604526405632385600541

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