Properties

Label 2-114-1.1-c5-0-15
Degree $2$
Conductor $114$
Sign $-1$
Analytic cond. $18.2837$
Root an. cond. $4.27595$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 9·3-s + 16·4-s − 91·5-s + 36·6-s − 33·7-s + 64·8-s + 81·9-s − 364·10-s − 91·11-s + 144·12-s − 610·13-s − 132·14-s − 819·15-s + 256·16-s − 1.83e3·17-s + 324·18-s − 361·19-s − 1.45e3·20-s − 297·21-s − 364·22-s − 3.43e3·23-s + 576·24-s + 5.15e3·25-s − 2.44e3·26-s + 729·27-s − 528·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.62·5-s + 0.408·6-s − 0.254·7-s + 0.353·8-s + 1/3·9-s − 1.15·10-s − 0.226·11-s + 0.288·12-s − 1.00·13-s − 0.179·14-s − 0.939·15-s + 1/4·16-s − 1.53·17-s + 0.235·18-s − 0.229·19-s − 0.813·20-s − 0.146·21-s − 0.160·22-s − 1.35·23-s + 0.204·24-s + 1.64·25-s − 0.707·26-s + 0.192·27-s − 0.127·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114\)    =    \(2 \cdot 3 \cdot 19\)
Sign: $-1$
Analytic conductor: \(18.2837\)
Root analytic conductor: \(4.27595\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 114,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p^{2} T \)
3 \( 1 - p^{2} T \)
19 \( 1 + p^{2} T \)
good5 \( 1 + 91 T + p^{5} T^{2} \)
7 \( 1 + 33 T + p^{5} T^{2} \)
11 \( 1 + 91 T + p^{5} T^{2} \)
13 \( 1 + 610 T + p^{5} T^{2} \)
17 \( 1 + 1833 T + p^{5} T^{2} \)
23 \( 1 + 3436 T + p^{5} T^{2} \)
29 \( 1 - 3562 T + p^{5} T^{2} \)
31 \( 1 - 322 T + p^{5} T^{2} \)
37 \( 1 - 7216 T + p^{5} T^{2} \)
41 \( 1 + 13664 T + p^{5} T^{2} \)
43 \( 1 + 3701 T + p^{5} T^{2} \)
47 \( 1 - 9203 T + p^{5} T^{2} \)
53 \( 1 - 29186 T + p^{5} T^{2} \)
59 \( 1 + 27804 T + p^{5} T^{2} \)
61 \( 1 - 707 p T + p^{5} T^{2} \)
67 \( 1 + 19428 T + p^{5} T^{2} \)
71 \( 1 - 7040 T + p^{5} T^{2} \)
73 \( 1 - 37341 T + p^{5} T^{2} \)
79 \( 1 + 4972 T + p^{5} T^{2} \)
83 \( 1 + 71196 T + p^{5} T^{2} \)
89 \( 1 + 3654 T + p^{5} T^{2} \)
97 \( 1 - 62362 T + p^{5} T^{2} \)
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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

−12.18154186201940451696645862947, −11.40665214775992343773469101076, −10.14901252213856620745307711736, −8.589247726013482235470834455842, −7.66204694811116789226974060714, −6.65969948246382348612165840012, −4.70678184402513013065539254176, −3.84156358943543872974532792638, −2.49591217082682398195681154641, 0, 2.49591217082682398195681154641, 3.84156358943543872974532792638, 4.70678184402513013065539254176, 6.65969948246382348612165840012, 7.66204694811116789226974060714, 8.589247726013482235470834455842, 10.14901252213856620745307711736, 11.40665214775992343773469101076, 12.18154186201940451696645862947

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