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Properties

Label 2-1216-19.7-c1-0-36
Degree $2$
Conductor $1216$
Sign $-0.980 + 0.194i$
Analytic cond. $9.70980$
Root an. cond. $3.11605$
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  + (0.5 + 0.866i)3-s + (−2 − 3.46i)5-s + (1 − 1.73i)9-s − 3·11-s + (1 − 1.73i)13-s + (1.99 − 3.46i)15-s + (−1 − 1.73i)17-s + (−0.5 + 4.33i)19-s + (−3 + 5.19i)23-s + (−5.49 + 9.52i)25-s + 5·27-s + (−2 + 3.46i)29-s − 10·31-s + (−1.5 − 2.59i)33-s − 2·37-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.894 − 1.54i)5-s + (0.333 − 0.577i)9-s − 0.904·11-s + (0.277 − 0.480i)13-s + (0.516 − 0.894i)15-s + (−0.242 − 0.420i)17-s + (−0.114 + 0.993i)19-s + (−0.625 + 1.08i)23-s + (−1.09 + 1.90i)25-s + 0.962·27-s + (−0.371 + 0.643i)29-s − 1.79·31-s + (−0.261 − 0.452i)33-s − 0.328·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.194i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 + 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $-0.980 + 0.194i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ -0.980 + 0.194i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4850743492\)
\(L(\frac12)\) \(\approx\) \(0.4850743492\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (0.5 - 4.33i)T \)
good3 \( 1 + (-0.5 - 0.866i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (2 + 3.46i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2 - 3.46i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 10T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + (4.5 + 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2 + 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6 + 10.3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1 - 1.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.5 + 7.79i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3 - 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.5 - 7.79i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 5T + 83T^{2} \)
89 \( 1 + (-9 + 15.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.194645270905816779697029808030, −8.627207374528811572045863895809, −7.88213138654611469930968810318, −7.14052847001835329244736612695, −5.56023829364467878995641976357, −5.12404206937091652157637415845, −3.92492264563539322797459055727, −3.53557239646990990256404256307, −1.66383371607342896001168565101, −0.19140794077643373687232908545, 2.07829031274344949334775058911, 2.85199342952786188851794984299, 3.89916287413110371298495766894, 4.88584097114635744909293674631, 6.35079965555136488708837351961, 6.83738908519454562562143914057, 7.82660739483147917959467981185, 7.998133238397861103457405460033, 9.279318677952284270476246114242, 10.42112824027713223787485408378

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