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Properties

Label 2-345-115.114-c2-0-12
Degree $2$
Conductor $345$
Sign $-0.989 - 0.143i$
Analytic cond. $9.40056$
Root an. cond. $3.06603$
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  + 2.58i·2-s + 1.73i·3-s − 2.65·4-s + (−2.43 − 4.36i)5-s − 4.46·6-s + 10.6·7-s + 3.45i·8-s − 2.99·9-s + (11.2 − 6.27i)10-s + 13.4i·11-s − 4.60i·12-s + 9.14i·13-s + 27.3i·14-s + (7.56 − 4.21i)15-s − 19.5·16-s − 13.0·17-s + ⋯
L(s)  = 1  + 1.29i·2-s + 0.577i·3-s − 0.664·4-s + (−0.486 − 0.873i)5-s − 0.744·6-s + 1.51·7-s + 0.432i·8-s − 0.333·9-s + (1.12 − 0.627i)10-s + 1.22i·11-s − 0.383i·12-s + 0.703i·13-s + 1.95i·14-s + (0.504 − 0.280i)15-s − 1.22·16-s − 0.765·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.143i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.989 - 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(345\)    =    \(3 \cdot 5 \cdot 23\)
Sign: $-0.989 - 0.143i$
Analytic conductor: \(9.40056\)
Root analytic conductor: \(3.06603\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{345} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 345,\ (\ :1),\ -0.989 - 0.143i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.115067 + 1.59791i\)
\(L(\frac12)\) \(\approx\) \(0.115067 + 1.59791i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.73iT \)
5 \( 1 + (2.43 + 4.36i)T \)
23 \( 1 + (-18.2 - 13.9i)T \)
good2 \( 1 - 2.58iT - 4T^{2} \)
7 \( 1 - 10.6T + 49T^{2} \)
11 \( 1 - 13.4iT - 121T^{2} \)
13 \( 1 - 9.14iT - 169T^{2} \)
17 \( 1 + 13.0T + 289T^{2} \)
19 \( 1 - 2.02iT - 361T^{2} \)
29 \( 1 + 51.3T + 841T^{2} \)
31 \( 1 - 40.4T + 961T^{2} \)
37 \( 1 + 21.4T + 1.36e3T^{2} \)
41 \( 1 - 63.6T + 1.68e3T^{2} \)
43 \( 1 - 52.8T + 1.84e3T^{2} \)
47 \( 1 - 26.1iT - 2.20e3T^{2} \)
53 \( 1 + 34.9T + 2.80e3T^{2} \)
59 \( 1 + 43.6T + 3.48e3T^{2} \)
61 \( 1 + 37.0iT - 3.72e3T^{2} \)
67 \( 1 + 11.7T + 4.48e3T^{2} \)
71 \( 1 + 28.0T + 5.04e3T^{2} \)
73 \( 1 - 69.7iT - 5.32e3T^{2} \)
79 \( 1 + 88.5iT - 6.24e3T^{2} \)
83 \( 1 + 17.5T + 6.88e3T^{2} \)
89 \( 1 + 4.49iT - 7.92e3T^{2} \)
97 \( 1 + 48.1T + 9.40e3T^{2} \)
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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

−11.51739482312741271327003215307, −11.06496872698906957679550214084, −9.412363897791244943080692892176, −8.788122787435911980392569915813, −7.80408868313787616750956218160, −7.24386761972312309456080693758, −5.75670998960489917717982652487, −4.71063250230832056537669909727, −4.41041363579153251816633494537, −1.87371661455241832581214815291, 0.75848350956810560044034343967, 2.21013210961587744817888012825, 3.19407486731318592108648141557, 4.44137055300206133582551700241, 5.94552170952761576108466237168, 7.19462880265839098657870380168, 8.069693206770403646873577387000, 9.005533603038129362386908613278, 10.54546963390550843106331726515, 11.10212036769496099410602732901

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