Properties

Label 2-156-52.51-c2-0-19
Degree $2$
Conductor $156$
Sign $-0.835 + 0.549i$
Analytic cond. $4.25069$
Root an. cond. $2.06172$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.31 − 1.50i)2-s − 1.73i·3-s + (−0.529 + 3.96i)4-s − 7.41i·5-s + (−2.60 + 2.28i)6-s + 13.3·7-s + (6.66 − 4.42i)8-s − 2.99·9-s + (−11.1 + 9.77i)10-s − 14.6·11-s + (6.86 + 0.917i)12-s + (9.82 − 8.51i)13-s + (−17.6 − 20.1i)14-s − 12.8·15-s + (−15.4 − 4.19i)16-s − 3.85·17-s + ⋯
L(s)  = 1  + (−0.658 − 0.752i)2-s − 0.577i·3-s + (−0.132 + 0.991i)4-s − 1.48i·5-s + (−0.434 + 0.380i)6-s + 1.91·7-s + (0.833 − 0.553i)8-s − 0.333·9-s + (−1.11 + 0.977i)10-s − 1.32·11-s + (0.572 + 0.0764i)12-s + (0.755 − 0.654i)13-s + (−1.25 − 1.43i)14-s − 0.856·15-s + (−0.964 − 0.262i)16-s − 0.226·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 + 0.549i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $-0.835 + 0.549i$
Analytic conductor: \(4.25069\)
Root analytic conductor: \(2.06172\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1),\ -0.835 + 0.549i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.312425 - 1.04457i\)
\(L(\frac12)\) \(\approx\) \(0.312425 - 1.04457i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.31 + 1.50i)T \)
3 \( 1 + 1.73iT \)
13 \( 1 + (-9.82 + 8.51i)T \)
good5 \( 1 + 7.41iT - 25T^{2} \)
7 \( 1 - 13.3T + 49T^{2} \)
11 \( 1 + 14.6T + 121T^{2} \)
17 \( 1 + 3.85T + 289T^{2} \)
19 \( 1 + 19.0T + 361T^{2} \)
23 \( 1 - 2.41iT - 529T^{2} \)
29 \( 1 - 18.9T + 841T^{2} \)
31 \( 1 + 11.2T + 961T^{2} \)
37 \( 1 + 38.0iT - 1.36e3T^{2} \)
41 \( 1 - 35.5iT - 1.68e3T^{2} \)
43 \( 1 - 10.1iT - 1.84e3T^{2} \)
47 \( 1 - 52.7T + 2.20e3T^{2} \)
53 \( 1 - 56.1T + 2.80e3T^{2} \)
59 \( 1 + 22.9T + 3.48e3T^{2} \)
61 \( 1 + 2.26T + 3.72e3T^{2} \)
67 \( 1 - 27.1T + 4.48e3T^{2} \)
71 \( 1 - 25.9T + 5.04e3T^{2} \)
73 \( 1 + 20.2iT - 5.32e3T^{2} \)
79 \( 1 + 14.9iT - 6.24e3T^{2} \)
83 \( 1 + 3.46T + 6.88e3T^{2} \)
89 \( 1 - 147. iT - 7.92e3T^{2} \)
97 \( 1 + 163. iT - 9.40e3T^{2} \)
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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

−12.31984678751275431445475437732, −11.24185979530062466012852706121, −10.53373385671965441302342437718, −8.838590531060455161096019540956, −8.283202901907801862097434510821, −7.67588247558374798332106363830, −5.40496703352416453769642292784, −4.41131242367626369999940589025, −2.13958070370836339479159814240, −0.910789158863463892228959989456, 2.21509371428643118439083554697, 4.42520740747789504138949769171, 5.61709025579703124402176102669, 6.91703684916580162143136984879, 7.966014086202873312155207083155, 8.736769698098825589693409926789, 10.41857052397747069208586129728, 10.72308957542980188990044438081, 11.50005004621093701057115048329, 13.71961513724455045927241384781

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