Properties

Label 2-26-13.12-c5-0-1
Degree $2$
Conductor $26$
Sign $-0.554 - 0.832i$
Analytic cond. $4.16997$
Root an. cond. $2.04205$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4i·2-s + 4·3-s − 16·4-s + 68i·5-s + 16i·6-s + 82i·7-s − 64i·8-s − 227·9-s − 272·10-s + 390i·11-s − 64·12-s + (507 − 338i)13-s − 328·14-s + 272i·15-s + 256·16-s + 1.73e3·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.256·3-s − 0.5·4-s + 1.21i·5-s + 0.181i·6-s + 0.632i·7-s − 0.353i·8-s − 0.934·9-s − 0.860·10-s + 0.971i·11-s − 0.128·12-s + (0.832 − 0.554i)13-s − 0.447·14-s + 0.312i·15-s + 0.250·16-s + 1.45·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(26\)    =    \(2 \cdot 13\)
Sign: $-0.554 - 0.832i$
Analytic conductor: \(4.16997\)
Root analytic conductor: \(2.04205\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{26} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 26,\ (\ :5/2),\ -0.554 - 0.832i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.650339 + 1.21516i\)
\(L(\frac12)\) \(\approx\) \(0.650339 + 1.21516i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 4iT \)
13 \( 1 + (-507 + 338i)T \)
good3 \( 1 - 4T + 243T^{2} \)
5 \( 1 - 68iT - 3.12e3T^{2} \)
7 \( 1 - 82iT - 1.68e4T^{2} \)
11 \( 1 - 390iT - 1.61e5T^{2} \)
17 \( 1 - 1.73e3T + 1.41e6T^{2} \)
19 \( 1 + 1.07e3iT - 2.47e6T^{2} \)
23 \( 1 - 2.10e3T + 6.43e6T^{2} \)
29 \( 1 + 1.69e3T + 2.05e7T^{2} \)
31 \( 1 + 1.43e3iT - 2.86e7T^{2} \)
37 \( 1 - 8.85e3iT - 6.93e7T^{2} \)
41 \( 1 - 6.76e3iT - 1.15e8T^{2} \)
43 \( 1 + 1.69e4T + 1.47e8T^{2} \)
47 \( 1 + 2.51e4iT - 2.29e8T^{2} \)
53 \( 1 - 3.82e4T + 4.18e8T^{2} \)
59 \( 1 - 2.12e4iT - 7.14e8T^{2} \)
61 \( 1 + 5.45e3T + 8.44e8T^{2} \)
67 \( 1 - 4.45e4iT - 1.35e9T^{2} \)
71 \( 1 + 1.77e4iT - 1.80e9T^{2} \)
73 \( 1 - 3.10e4iT - 2.07e9T^{2} \)
79 \( 1 + 4.53e4T + 3.07e9T^{2} \)
83 \( 1 + 1.24e5iT - 3.93e9T^{2} \)
89 \( 1 + 1.87e4iT - 5.58e9T^{2} \)
97 \( 1 + 1.21e5iT - 8.58e9T^{2} \)
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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

−16.84422693756525128178896545524, −15.13920759107341829461285657612, −14.81394533840619131101820221798, −13.41176053408429832643652919933, −11.69152799290617391231480299779, −10.14478618619664159513771230615, −8.556989399334337280439412409777, −7.09893731325060923102133084336, −5.61533935031378376287663561628, −3.06335241862487633024767743322, 0.997819899314083278539394913265, 3.61417921659398664475238981520, 5.52496987888834141743494460074, 8.199264892661609685839741617723, 9.182734194242442678162213371683, 10.87235732320394567618449246828, 12.12896042132615130489689778417, 13.42023699039455229758313516674, 14.32030532546583843582309130801, 16.41800088319794670756348266335

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