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Properties

Label 2-575-115.22-c1-0-13
Degree $2$
Conductor $575$
Sign $-0.417 - 0.908i$
Analytic cond. $4.59139$
Root an. cond. $2.14275$
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  + (1.22 + 1.22i)2-s + 0.999i·4-s + (−2.34 + 2.34i)7-s + (1.22 − 1.22i)8-s + 3i·9-s + 5.74i·11-s + (1.22 − 1.22i)13-s − 5.74·14-s + 5·16-s + (4.69 − 4.69i)17-s + (−3.67 + 3.67i)18-s − 5.74·19-s + (−7.03 + 7.03i)22-s + (−0.104 + 4.79i)23-s + 2.99·26-s + ⋯
L(s)  = 1  + (0.866 + 0.866i)2-s + 0.499i·4-s + (−0.886 + 0.886i)7-s + (0.433 − 0.433i)8-s + i·9-s + 1.73i·11-s + (0.339 − 0.339i)13-s − 1.53·14-s + 1.25·16-s + (1.13 − 1.13i)17-s + (−0.866 + 0.866i)18-s − 1.31·19-s + (−1.49 + 1.49i)22-s + (−0.0217 + 0.999i)23-s + 0.588·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.417 - 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.417 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(575\)    =    \(5^{2} \cdot 23\)
Sign: $-0.417 - 0.908i$
Analytic conductor: \(4.59139\)
Root analytic conductor: \(2.14275\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{575} (482, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 575,\ (\ :1/2),\ -0.417 - 0.908i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12944 + 1.76278i\)
\(L(\frac12)\) \(\approx\) \(1.12944 + 1.76278i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
23 \( 1 + (0.104 - 4.79i)T \)
good2 \( 1 + (-1.22 - 1.22i)T + 2iT^{2} \)
3 \( 1 - 3iT^{2} \)
7 \( 1 + (2.34 - 2.34i)T - 7iT^{2} \)
11 \( 1 - 5.74iT - 11T^{2} \)
13 \( 1 + (-1.22 + 1.22i)T - 13iT^{2} \)
17 \( 1 + (-4.69 + 4.69i)T - 17iT^{2} \)
19 \( 1 + 5.74T + 19T^{2} \)
29 \( 1 - 7iT - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (-4.69 + 4.69i)T - 37iT^{2} \)
41 \( 1 - 7T + 41T^{2} \)
43 \( 1 + (2.34 + 2.34i)T + 43iT^{2} \)
47 \( 1 + (7.34 + 7.34i)T + 47iT^{2} \)
53 \( 1 + 53iT^{2} \)
59 \( 1 + 10iT - 59T^{2} \)
61 \( 1 + 11.4iT - 61T^{2} \)
67 \( 1 + (-4.69 + 4.69i)T - 67iT^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + (-1.22 + 1.22i)T - 73iT^{2} \)
79 \( 1 - 5.74T + 79T^{2} \)
83 \( 1 + (-7.03 - 7.03i)T + 83iT^{2} \)
89 \( 1 - 11.4T + 89T^{2} \)
97 \( 1 + (4.69 - 4.69i)T - 97iT^{2} \)
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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

−10.98286813055871963076592571260, −9.964759544515021426953087937256, −9.394640596043954300551761896271, −7.987399526689871951764828272994, −7.23865285457262218778671518673, −6.40728181960222315455490180709, −5.36146370298640918605656596345, −4.84708867992634175717696410568, −3.50904128288593569267543119226, −2.09368323811470264015678002459, 0.956722660437102283840609278946, 2.84823454353044133134567823076, 3.70732574964921737754288430641, 4.24442121833120461196938951553, 6.02539430278843490353674099604, 6.32816681351103571906698278180, 7.907825969257948427028115917814, 8.691234273601879663602813770143, 9.945152211216240564321762942087, 10.63277133315145323500037254670

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