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Properties

Label 2-39-13.3-c1-0-1
Degree $2$
Conductor $39$
Sign $0.794 - 0.607i$
Analytic cond. $0.311416$
Root an. cond. $0.558047$
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  + (0.780 + 1.35i)2-s + (−0.5 − 0.866i)3-s + (−0.219 + 0.379i)4-s − 3.56·5-s + (0.780 − 1.35i)6-s + (−0.280 + 0.486i)7-s + 2.43·8-s + (−0.499 + 0.866i)9-s + (−2.78 − 4.81i)10-s + (1 + 1.73i)11-s + 0.438·12-s + (0.5 − 3.57i)13-s − 0.876·14-s + (1.78 + 3.08i)15-s + (2.34 + 4.05i)16-s + (0.780 − 1.35i)17-s + ⋯
L(s)  = 1  + (0.552 + 0.956i)2-s + (−0.288 − 0.499i)3-s + (−0.109 + 0.189i)4-s − 1.59·5-s + (0.318 − 0.552i)6-s + (−0.106 + 0.183i)7-s + 0.862·8-s + (−0.166 + 0.288i)9-s + (−0.879 − 1.52i)10-s + (0.301 + 0.522i)11-s + 0.126·12-s + (0.138 − 0.990i)13-s − 0.234·14-s + (0.459 + 0.796i)15-s + (0.585 + 1.01i)16-s + (0.189 − 0.327i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 - 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.794 - 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.794 - 0.607i$
Analytic conductor: \(0.311416\)
Root analytic conductor: \(0.558047\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :1/2),\ 0.794 - 0.607i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.772749 + 0.261532i\)
\(L(\frac12)\) \(\approx\) \(0.772749 + 0.261532i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.5 + 3.57i)T \)
good2 \( 1 + (-0.780 - 1.35i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 3.56T + 5T^{2} \)
7 \( 1 + (0.280 - 0.486i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.780 + 1.35i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.56 - 6.16i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1 + 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.34 + 5.78i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.56T + 31T^{2} \)
37 \( 1 + (3.78 + 6.54i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.780 - 1.35i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.28 - 3.95i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 8.24T + 47T^{2} \)
53 \( 1 + 0.684T + 53T^{2} \)
59 \( 1 + (-1.43 + 2.49i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.93 - 3.35i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.28 + 3.95i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (7 - 12.1i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 10.1T + 73T^{2} \)
79 \( 1 - 5.43T + 79T^{2} \)
83 \( 1 + 0.876T + 83T^{2} \)
89 \( 1 + (2.43 + 4.22i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.28 + 7.41i)T + (-48.5 - 84.0i)T^{2} \)
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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

−16.10483765269793869273499649411, −15.30897100800727125543660371756, −14.43100428230104194527936293766, −12.85671032012627204036292872112, −11.91949977225508773089406983661, −10.54903101242059444721771061579, −8.136387494415144605340527017090, −7.34184704771923900614729101156, −5.89823560273817359649723768672, −4.18456692599948645079505875420, 3.52701343845126251655004938817, 4.53128171992419323429389545167, 7.08999134128220321789971983341, 8.718554065492295159662662375559, 10.66784836250385217406801053823, 11.46767029879749265396053963398, 12.19312672416011351645930905074, 13.59148788467761498803386557713, 15.06129333520377956548833961225, 16.13538705210786371922051987182

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