Properties

Label 2-1450-5.4-c1-0-30
Degree $2$
Conductor $1450$
Sign $-0.894 + 0.447i$
Analytic cond. $11.5783$
Root an. cond. $3.40269$
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  i·2-s i·3-s − 4-s − 6-s + 2i·7-s + i·8-s + 2·9-s − 3·11-s + i·12-s i·13-s + 2·14-s + 16-s − 8i·17-s − 2i·18-s + 2·21-s + 3i·22-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s + 0.755i·7-s + 0.353i·8-s + 0.666·9-s − 0.904·11-s + 0.288i·12-s − 0.277i·13-s + 0.534·14-s + 0.250·16-s − 1.94i·17-s − 0.471i·18-s + 0.436·21-s + 0.639i·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1450\)    =    \(2 \cdot 5^{2} \cdot 29\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(11.5783\)
Root analytic conductor: \(3.40269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1450} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1450,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.222811854\)
\(L(\frac12)\) \(\approx\) \(1.222811854\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 \)
29 \( 1 - T \)
good3 \( 1 + iT - 3T^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + iT - 13T^{2} \)
17 \( 1 + 8iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
31 \( 1 + 3T + 31T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 11iT - 43T^{2} \)
47 \( 1 + 13iT - 47T^{2} \)
53 \( 1 + 11iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 + 15T + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−9.282708863658135757727897311368, −8.487515671598840071247803635051, −7.48402816835388469244948131603, −7.01138723621797215961796653114, −5.56138675528115967421074827793, −5.14278588832138914268892894165, −3.86934664072090993796106148388, −2.72117007321105440843291562410, −1.99883379181252830533277575355, −0.50934430537415501617057669376, 1.43337419471930793188795317925, 3.13815152938881594795222410751, 4.31172238911361323212801375920, 4.58615571580410205780919613967, 5.87624576083343006235451222415, 6.55377279749605764139928304380, 7.57394862267748566163786555293, 8.074362155823193822037270074564, 9.041531891321149447817572604306, 9.885830763888100270512883690511

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