Properties

Label 2-1-1.1-c25-0-0
Degree $2$
Conductor $1$
Sign $-1$
Analytic cond. $3.95996$
Root an. cond. $1.98996$
Motivic weight $25$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 48·2-s − 1.95e5·3-s − 3.35e7·4-s − 7.41e8·5-s + 9.39e6·6-s + 3.90e10·7-s + 3.22e9·8-s − 8.08e11·9-s + 3.56e10·10-s + 8.41e12·11-s + 6.56e12·12-s − 8.16e13·13-s − 1.87e12·14-s + 1.45e14·15-s + 1.12e15·16-s − 2.51e15·17-s + 3.88e13·18-s − 6.08e15·19-s + 2.48e16·20-s − 7.65e15·21-s − 4.04e14·22-s − 9.49e16·23-s − 6.30e14·24-s + 2.52e17·25-s + 3.91e15·26-s + 3.24e17·27-s − 1.31e18·28-s + ⋯
L(s)  = 1  − 0.00828·2-s − 0.212·3-s − 0.999·4-s − 1.35·5-s + 0.00176·6-s + 1.06·7-s + 0.0165·8-s − 0.954·9-s + 0.0112·10-s + 0.808·11-s + 0.212·12-s − 0.972·13-s − 0.00884·14-s + 0.289·15-s + 0.999·16-s − 1.04·17-s + 0.00791·18-s − 0.630·19-s + 1.35·20-s − 0.227·21-s − 0.00670·22-s − 0.903·23-s − 0.00352·24-s + 0.847·25-s + 0.00805·26-s + 0.415·27-s − 1.06·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(26-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+25/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $-1$
Analytic conductor: \(3.95996\)
Root analytic conductor: \(1.98996\)
Motivic weight: \(25\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1,\ (\ :25/2),\ -1)\)

Particular Values

\(L(13)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{27}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 + 3 p^{4} T + p^{25} T^{2} \)
3 \( 1 + 7252 p^{3} T + p^{25} T^{2} \)
5 \( 1 + 29679594 p^{2} T + p^{25} T^{2} \)
7 \( 1 - 797563208 p^{2} T + p^{25} T^{2} \)
11 \( 1 - 765410481732 p T + p^{25} T^{2} \)
13 \( 1 + 6280849641178 p T + p^{25} T^{2} \)
17 \( 1 + 148229413467534 p T + p^{25} T^{2} \)
19 \( 1 + 320108230016260 p T + p^{25} T^{2} \)
23 \( 1 + 4130229578100888 p T + p^{25} T^{2} \)
29 \( 1 + 271246959476737410 T + p^{25} T^{2} \)
31 \( 1 - 4291666067521509152 T + p^{25} T^{2} \)
37 \( 1 - 20301484446109126982 T + p^{25} T^{2} \)
41 \( 1 + \)\(18\!\cdots\!98\)\( T + p^{25} T^{2} \)
43 \( 1 - \)\(30\!\cdots\!56\)\( T + p^{25} T^{2} \)
47 \( 1 + \)\(92\!\cdots\!88\)\( T + p^{25} T^{2} \)
53 \( 1 + \)\(99\!\cdots\!54\)\( T + p^{25} T^{2} \)
59 \( 1 - \)\(13\!\cdots\!80\)\( T + p^{25} T^{2} \)
61 \( 1 - \)\(90\!\cdots\!02\)\( T + p^{25} T^{2} \)
67 \( 1 + \)\(26\!\cdots\!28\)\( T + p^{25} T^{2} \)
71 \( 1 + \)\(19\!\cdots\!48\)\( T + p^{25} T^{2} \)
73 \( 1 - \)\(42\!\cdots\!26\)\( T + p^{25} T^{2} \)
79 \( 1 + \)\(27\!\cdots\!60\)\( T + p^{25} T^{2} \)
83 \( 1 + \)\(93\!\cdots\!84\)\( T + p^{25} T^{2} \)
89 \( 1 + \)\(17\!\cdots\!30\)\( T + p^{25} T^{2} \)
97 \( 1 - \)\(28\!\cdots\!62\)\( T + p^{25} T^{2} \)
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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

−26.88537206297050000566551198613, −23.93312251573463892822942793783, −22.42268203005764508318554782850, −19.62814349222831481270313941415, −17.43082052812684484241584138314, −14.61160856189401032583901840016, −11.70089941202410050445650230556, −8.332583170673278606893499008029, −4.43532131875266390947498555989, 0, 4.43532131875266390947498555989, 8.332583170673278606893499008029, 11.70089941202410050445650230556, 14.61160856189401032583901840016, 17.43082052812684484241584138314, 19.62814349222831481270313941415, 22.42268203005764508318554782850, 23.93312251573463892822942793783, 26.88537206297050000566551198613

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