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Properties

Label 2-5e2-5.4-c33-0-3
Degree $2$
Conductor $25$
Sign $0.447 + 0.894i$
Analytic cond. $172.457$
Root an. cond. $13.1322$
Motivic weight $33$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.29e5i·2-s − 6.82e7i·3-s − 8.07e9·4-s + 8.81e12·6-s + 1.39e14i·7-s + 6.68e13i·8-s + 8.95e14·9-s − 1.11e17·11-s + 5.51e17i·12-s + 3.82e18i·13-s − 1.79e19·14-s − 7.79e19·16-s − 1.35e20i·17-s + 1.15e20i·18-s − 1.50e21·19-s + ⋯
L(s)  = 1  + 1.39i·2-s − 0.915i·3-s − 0.939·4-s + 1.27·6-s + 1.58i·7-s + 0.0839i·8-s + 0.161·9-s − 0.733·11-s + 0.860i·12-s + 1.59i·13-s − 2.20·14-s − 1.05·16-s − 0.676i·17-s + 0.224i·18-s − 1.20·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(172.457\)
Root analytic conductor: \(13.1322\)
Motivic weight: \(33\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :33/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(17)\) \(\approx\) \(0.4770646747\)
\(L(\frac12)\) \(\approx\) \(0.4770646747\)
\(L(\frac{35}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 - 1.29e5iT - 8.58e9T^{2} \)
3 \( 1 + 6.82e7iT - 5.55e15T^{2} \)
7 \( 1 - 1.39e14iT - 7.73e27T^{2} \)
11 \( 1 + 1.11e17T + 2.32e34T^{2} \)
13 \( 1 - 3.82e18iT - 5.75e36T^{2} \)
17 \( 1 + 1.35e20iT - 4.02e40T^{2} \)
19 \( 1 + 1.50e21T + 1.58e42T^{2} \)
23 \( 1 - 2.26e22iT - 8.65e44T^{2} \)
29 \( 1 - 6.82e23T + 1.81e48T^{2} \)
31 \( 1 + 5.41e24T + 1.64e49T^{2} \)
37 \( 1 - 7.55e24iT - 5.63e51T^{2} \)
41 \( 1 + 1.95e26T + 1.66e53T^{2} \)
43 \( 1 - 5.44e26iT - 8.02e53T^{2} \)
47 \( 1 - 3.58e27iT - 1.51e55T^{2} \)
53 \( 1 - 4.28e28iT - 7.96e56T^{2} \)
59 \( 1 - 2.15e28T + 2.74e58T^{2} \)
61 \( 1 - 5.24e29T + 8.23e58T^{2} \)
67 \( 1 - 1.47e30iT - 1.82e60T^{2} \)
71 \( 1 + 1.24e30T + 1.23e61T^{2} \)
73 \( 1 + 9.14e30iT - 3.08e61T^{2} \)
79 \( 1 - 2.52e31T + 4.18e62T^{2} \)
83 \( 1 + 1.28e31iT - 2.13e63T^{2} \)
89 \( 1 + 1.22e32T + 2.13e64T^{2} \)
97 \( 1 + 2.05e31iT - 3.65e65T^{2} \)
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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

−12.42331307198519747521979876446, −11.39937596509654360946205779946, −9.324552887713666795389488146493, −8.464680453650848655171553743471, −7.38814876276216471968003615540, −6.52985772262361883162884027652, −5.67692559773676671344757167309, −4.57169764010908605708652284510, −2.48242534172633457939013940832, −1.76114186697325685428212704680, 0.10264365228404228580602590691, 0.843304394814947824292199290077, 2.13530817231105033458668058158, 3.46622149353735220547283791995, 3.99071866224423673391457295531, 5.07263920681015168828072867687, 6.93506225703746271152699750144, 8.324619991084214599887509209862, 9.966861057859193455262177118254, 10.48370481306758051620181674438

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