Properties

Label 2-109-109.38-c1-0-0
Degree $2$
Conductor $109$
Sign $-0.943 + 0.331i$
Analytic cond. $0.870369$
Root an. cond. $0.932935$
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  + (−1.09 + 1.90i)2-s + (−0.271 + 1.53i)3-s + (−1.41 − 2.45i)4-s + (−1.85 + 0.676i)5-s + (−2.63 − 2.20i)6-s + (−0.0942 + 0.0343i)7-s + 1.83·8-s + (0.526 + 0.191i)9-s + (0.754 − 4.28i)10-s + (0.0390 + 0.221i)11-s + (4.15 − 1.51i)12-s + (−0.595 + 0.216i)13-s + (0.0383 − 0.217i)14-s + (−0.536 − 3.04i)15-s + (0.818 − 1.41i)16-s + (−2.64 + 4.58i)17-s + ⋯
L(s)  = 1  + (−0.777 + 1.34i)2-s + (−0.156 + 0.888i)3-s + (−0.708 − 1.22i)4-s + (−0.830 + 0.302i)5-s + (−1.07 − 0.901i)6-s + (−0.0356 + 0.0129i)7-s + 0.647·8-s + (0.175 + 0.0638i)9-s + (0.238 − 1.35i)10-s + (0.0117 + 0.0668i)11-s + (1.20 − 0.436i)12-s + (−0.165 + 0.0601i)13-s + (0.0102 − 0.0580i)14-s + (−0.138 − 0.785i)15-s + (0.204 − 0.354i)16-s + (−0.642 + 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.331i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.943 + 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(109\)
Sign: $-0.943 + 0.331i$
Analytic conductor: \(0.870369\)
Root analytic conductor: \(0.932935\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{109} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 109,\ (\ :1/2),\ -0.943 + 0.331i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0854609 - 0.501372i\)
\(L(\frac12)\) \(\approx\) \(0.0854609 - 0.501372i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad109 \( 1 + (8.93 + 5.40i)T \)
good2 \( 1 + (1.09 - 1.90i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (0.271 - 1.53i)T + (-2.81 - 1.02i)T^{2} \)
5 \( 1 + (1.85 - 0.676i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (0.0942 - 0.0343i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (-0.0390 - 0.221i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (0.595 - 0.216i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (2.64 - 4.58i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0578 - 0.100i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.60 - 6.24i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.712 + 4.04i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-9.42 - 3.43i)T + (23.7 + 19.9i)T^{2} \)
37 \( 1 + (3.01 + 1.09i)T + (28.3 + 23.7i)T^{2} \)
41 \( 1 + 3.58T + 41T^{2} \)
43 \( 1 + (-6.02 + 10.4i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.16 + 3.49i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (-10.2 + 3.71i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (1.29 + 7.36i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-4.67 - 3.92i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-2.40 - 0.875i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (3.43 + 5.94i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.571 - 3.24i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (-12.7 - 4.62i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-1.05 + 5.97i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (-6.50 - 5.45i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (15.4 - 5.62i)T + (74.3 - 62.3i)T^{2} \)
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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

−15.00851513664956572802759851820, −13.59055807233624957438718904015, −11.99550407095698153473531775790, −10.75786070613133534747654970189, −9.806097234447687781144036792025, −8.732788308876264028582643061125, −7.68472301474050460728225174052, −6.70749793111622137224726154925, −5.27688204579248591201961832485, −3.86812673419520033835565475773, 0.78259659388587214092145204096, 2.64952853342282986332259066585, 4.40189748754000576834851652476, 6.65057963660428959678099546439, 7.88412970661048320809559487754, 8.875113174585207800416239017334, 10.04772173474360175538489837124, 11.22126972283372986415791561619, 11.99649488777369815614179253649, 12.62221899226309365307908665981

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