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Properties

Label 2-112-112.37-c1-0-0
Degree $2$
Conductor $112$
Sign $-0.789 - 0.613i$
Analytic cond. $0.894324$
Root an. cond. $0.945687$
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  + (−1.40 + 0.164i)2-s + (−0.839 − 0.224i)3-s + (1.94 − 0.461i)4-s + (−3.16 + 0.847i)5-s + (1.21 + 0.177i)6-s + (0.654 + 2.56i)7-s + (−2.65 + 0.968i)8-s + (−1.94 − 1.12i)9-s + (4.30 − 1.71i)10-s + (−0.769 + 2.87i)11-s + (−1.73 − 0.0499i)12-s + (−3.63 + 3.63i)13-s + (−1.34 − 3.49i)14-s + 2.84·15-s + (3.57 − 1.79i)16-s + (−1.81 − 3.14i)17-s + ⋯
L(s)  = 1  + (−0.993 + 0.116i)2-s + (−0.484 − 0.129i)3-s + (0.972 − 0.230i)4-s + (−1.41 + 0.379i)5-s + (0.496 + 0.0726i)6-s + (0.247 + 0.968i)7-s + (−0.939 + 0.342i)8-s + (−0.648 − 0.374i)9-s + (1.36 − 0.540i)10-s + (−0.231 + 0.865i)11-s + (−0.501 − 0.0144i)12-s + (−1.00 + 1.00i)13-s + (−0.358 − 0.933i)14-s + 0.734·15-s + (0.893 − 0.449i)16-s + (−0.441 − 0.763i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.789 - 0.613i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.789 - 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $-0.789 - 0.613i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :1/2),\ -0.789 - 0.613i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0753860 + 0.219980i\)
\(L(\frac12)\) \(\approx\) \(0.0753860 + 0.219980i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 - 0.164i)T \)
7 \( 1 + (-0.654 - 2.56i)T \)
good3 \( 1 + (0.839 + 0.224i)T + (2.59 + 1.5i)T^{2} \)
5 \( 1 + (3.16 - 0.847i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.769 - 2.87i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (3.63 - 3.63i)T - 13iT^{2} \)
17 \( 1 + (1.81 + 3.14i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.429 + 1.60i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-5.33 - 3.08i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.10 + 5.10i)T - 29iT^{2} \)
31 \( 1 + (1.00 + 1.74i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.57 - 1.49i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 3.71iT - 41T^{2} \)
43 \( 1 + (2.91 + 2.91i)T + 43iT^{2} \)
47 \( 1 + (5.06 - 8.77i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.986 - 3.68i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-0.977 + 3.64i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.75 - 6.54i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-5.88 - 1.57i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 9.55iT - 71T^{2} \)
73 \( 1 + (-0.989 + 0.571i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.120 + 0.209i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.459 - 0.459i)T - 83iT^{2} \)
89 \( 1 + (3.76 + 2.17i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 6.80T + 97T^{2} \)
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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

−14.55286627357507557068617551885, −12.34880492858744085125669241657, −11.65811117911733303294938361417, −11.25678926102441625620739766947, −9.635330298337427061603555778720, −8.656387554575693142841787173189, −7.46564465352441808648658817615, −6.64234852843095132066525244611, −4.93778882683667742399381127131, −2.70018096490040559913162750230, 0.35536607143618781339084826233, 3.34059042901607102705512413560, 5.06874256256644158814149320930, 6.86975382053626399233080118077, 8.027989461570045819804574107778, 8.538848752283682689791256405118, 10.45501937540091245393199021902, 10.88555133880445192128073890917, 11.87770620095612682207949170142, 12.82100170742981068292507611327

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