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Properties

Label 2-1575-105.89-c1-0-31
Degree $2$
Conductor $1575$
Sign $0.673 + 0.739i$
Analytic cond. $12.5764$
Root an. cond. $3.54632$
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  + (0.707 − 1.22i)2-s + (2.59 − 0.5i)7-s + 2.82·8-s + (1.22 − 0.707i)11-s + 5.19·13-s + (1.22 − 3.53i)14-s + (2.00 − 3.46i)16-s + (−4.24 + 2.44i)17-s + (−1.5 − 0.866i)19-s − 2i·22-s + (−2.82 + 4.89i)23-s + (3.67 − 6.36i)26-s + 2.82i·29-s + (1.5 − 0.866i)31-s + 6.92i·34-s + ⋯
L(s)  = 1  + (0.499 − 0.866i)2-s + (0.981 − 0.188i)7-s + 0.999·8-s + (0.369 − 0.213i)11-s + 1.44·13-s + (0.327 − 0.944i)14-s + (0.500 − 0.866i)16-s + (−1.02 + 0.594i)17-s + (−0.344 − 0.198i)19-s − 0.426i·22-s + (−0.589 + 1.02i)23-s + (0.720 − 1.24i)26-s + 0.525i·29-s + (0.269 − 0.155i)31-s + 1.18i·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.673 + 0.739i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.673 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.673 + 0.739i$
Analytic conductor: \(12.5764\)
Root analytic conductor: \(3.54632\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :1/2),\ 0.673 + 0.739i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.983485358\)
\(L(\frac12)\) \(\approx\) \(2.983485358\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 + (-2.59 + 0.5i)T \)
good2 \( 1 + (-0.707 + 1.22i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (-1.22 + 0.707i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.19T + 13T^{2} \)
17 \( 1 + (4.24 - 2.44i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.5 + 0.866i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.82 - 4.89i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.82iT - 29T^{2} \)
31 \( 1 + (-1.5 + 0.866i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.866 - 0.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.34T + 41T^{2} \)
43 \( 1 - iT - 43T^{2} \)
47 \( 1 + (10.6 + 6.12i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.41 - 2.44i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.44 + 4.24i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3 + 1.73i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.52 + 5.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.07iT - 71T^{2} \)
73 \( 1 + (0.866 + 1.5i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.5 + 4.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.34iT - 83T^{2} \)
89 \( 1 + (-2.44 + 4.24i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.3T + 97T^{2} \)
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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

−9.369127124651524261752716524091, −8.417193982242333056275330075723, −7.938352391432324949026041135247, −6.86271970355010446259285869510, −5.98304494642646902406674106885, −4.86297695492758888401545692228, −4.06982155015789405412873777398, −3.43368533304455412826007418105, −2.09521962555785351838246579840, −1.32535572009225789476010591542, 1.27959434471471848582613754179, 2.37370319368140202659082663126, 4.07440184230239746732693232828, 4.54393590886909468788834681516, 5.55286634234359397674562096191, 6.29940297318426883002403566484, 6.88309697444528063720403480730, 7.968385906079358627880104072246, 8.426698783919758535323739839473, 9.364971052146766835817194115681

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