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Properties

Label 2-6300-21.5-c1-0-0
Degree $2$
Conductor $6300$
Sign $-0.774 + 0.632i$
Analytic cond. $50.3057$
Root an. cond. $7.09265$
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  + (−2.61 − 0.380i)7-s + (−2.15 + 1.24i)11-s + 6.80i·13-s + (3.90 + 6.75i)17-s + (−2.13 − 1.23i)19-s + (−3.14 − 1.81i)23-s + 2.55i·29-s + (3.68 − 2.12i)31-s + (0.925 − 1.60i)37-s − 4.23·41-s − 8.70·43-s + (0.516 − 0.894i)47-s + (6.71 + 1.99i)49-s + (−0.0496 + 0.0286i)53-s + (−5.59 − 9.68i)59-s + ⋯
L(s)  = 1  + (−0.989 − 0.143i)7-s + (−0.650 + 0.375i)11-s + 1.88i·13-s + (0.946 + 1.63i)17-s + (−0.489 − 0.282i)19-s + (−0.654 − 0.378i)23-s + 0.473i·29-s + (0.662 − 0.382i)31-s + (0.152 − 0.263i)37-s − 0.661·41-s − 1.32·43-s + (0.0752 − 0.130i)47-s + (0.958 + 0.284i)49-s + (−0.00681 + 0.00393i)53-s + (−0.728 − 1.26i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.774 + 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6300\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.774 + 0.632i$
Analytic conductor: \(50.3057\)
Root analytic conductor: \(7.09265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{6300} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6300,\ (\ :1/2),\ -0.774 + 0.632i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.09443537344\)
\(L(\frac12)\) \(\approx\) \(0.09443537344\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (2.61 + 0.380i)T \)
good11 \( 1 + (2.15 - 1.24i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 6.80iT - 13T^{2} \)
17 \( 1 + (-3.90 - 6.75i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.13 + 1.23i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.14 + 1.81i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.55iT - 29T^{2} \)
31 \( 1 + (-3.68 + 2.12i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.925 + 1.60i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.23T + 41T^{2} \)
43 \( 1 + 8.70T + 43T^{2} \)
47 \( 1 + (-0.516 + 0.894i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.0496 - 0.0286i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.59 + 9.68i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.69 - 1.55i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.18 - 8.97i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.13iT - 71T^{2} \)
73 \( 1 + (3.59 - 2.07i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.08 + 3.61i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.7T + 83T^{2} \)
89 \( 1 + (-4.98 + 8.62i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 4.91iT - 97T^{2} \)
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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

−8.425128221379490224508782197426, −7.82147907192519834008467482443, −6.81955854734443053757133010149, −6.51139631702790555373835064848, −5.77503539122763511438169663355, −4.78008705526161694914684230177, −4.04610623326721901457963585720, −3.42517897218758725321688835502, −2.32509083963118856337693298761, −1.56869629857939768888662937584, 0.02729036895559890595693883771, 0.938210526546038063433114800101, 2.52504571936755498785403187696, 3.07042673225503551147259321095, 3.67197589748364945001242182150, 4.97496869360090993993544694214, 5.44915345697569536768136899546, 6.12779195227613303502228959601, 6.88502984549966602997379615278, 7.84988420612258705739088848972

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